Felix' Math Place (Posts about Category Theory.)
https://math.fontein.de/category/category-theory.atom
2019-11-17T10:38:26Z
Felix Fontein
Nikola
Fun With Representable Functors, or Why I Like Yondea's Lemma.
https://math.fontein.de/2009/08/16/fun-with-representable-functors-or-why-i-like-yondeas-lemma/
2009-08-16T19:53:38+02:00
2009-08-16T19:53:38+02:00
Felix Fontein
<div>
<p>
Sometimes people ask, “why do you use <a href="https://en.wikipedia.org/wiki/Category_theory">Category Theorey</a>? Isn't it just a set of abstract terms making things look more complicated?” Well, sometimes this is true. Often, it allows to make short and precise statements instead of listing several properties:
</p>
<blockquote class="block-quote">
<p>
“<span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/YONOJBogAF6CR7yxQzTqbuid2nML9DLXhxZHTA.svgz" alt="F" title="F"></span> is a functor from the category of groups into the category of sets” (or “<span class="inline-formula"><img class="img-inline-formula img-formula" width="101" height="16" src="https://math.fontein.de/formulae/4c445psWzA8_D7u2EEu4Zka.4Gxi7cjCFQzbrQ.svgz" alt="F : \catGrp \to \catSet" title="F : \catGrp \to \catSet"></span> is a functor”)
</p>
</blockquote>
<p>
versus
</p>
<blockquote class="block-quote">
<p>
“for every group, <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="18" src="https://math.fontein.de/formulae/IXoSlWlSngfwlyAK3cltY09MehJ9ECr.jNGlNA.svgz" alt="F(G)" title="F(G)"></span> is a set; if <span class="inline-formula"><img class="img-inline-formula img-formula" width="86" height="17" src="https://math.fontein.de/formulae/7Wy9USGlMrc2VIvRrUCkE4ijv8dy.WshoDrVaA.svgz" alt="\phi : G \to G'" title="\phi : G \to G'"></span> is a group homomorphism, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="169" height="18" src="https://math.fontein.de/formulae/YvHoLYWdxnwRlqdpsU0spsCphFktb7pg7k7Xww.svgz" alt="F(\phi) : F(G) \to F(G')" title="F(\phi) : F(G) \to F(G')"></span> is a map; if <span class="inline-formula"><img class="img-inline-formula img-formula" width="61" height="16" src="https://math.fontein.de/formulae/5VCoG9BUTJgDAnkTz.FeI_VAjJuW6XXddnBCSg.svgz" alt="\phi = \id_G" title="\phi = \id_G"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="110" height="20" src="https://math.fontein.de/formulae/EdU2wpRbhcb9c9oC2eTyxy4Of53iuAtXJYqopQ.svgz" alt="F(\phi) = \id_{F(G)}" title="F(\phi) = \id_{F(G)}"></span>; if <span class="inline-formula"><img class="img-inline-formula img-formula" width="96" height="17" src="https://math.fontein.de/formulae/eXCjQVNo9DSEhy9JbsdyNnjOLSfibMzT.zwB2w.svgz" alt="\psi : G' \to G''" title="\psi : G' \to G''"></span> is another group homomorphism, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="185" height="18" src="https://math.fontein.de/formulae/pu8PwSpt8GiLGuHkKWiRIx0R1ZhpN5jCg7FTmA.svgz" alt="F(\psi \circ \phi) = F(\psi) \circ F(\phi)" title="F(\psi \circ \phi) = F(\psi) \circ F(\phi)"></span>”.
</p>
</blockquote>
<p>
And it allows to apply very generic statements to a large class of specific examples. And, sometimes, it even gives new insights by abstracting results.
</p>
<p>
Today I want to discuss representable functors and Yondea's lemma which is, for example, used a lot in modern <a href="https://en.wikipedia.org/wiki/Algebraic_geometry">Algebraic Geometry</a>. In the following, I will always assume that in all categories I use, the morphisms between two objects form a set. We will denote the category of sets by <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="12" src="https://math.fontein.de/formulae/VOrl7F25VqQHv35P164VNh2Ckbi4H3WEeW2xrg.svgz" alt="\catSet" title="\catSet"></span> (and not the french <span class="inline-formula"><img class="img-inline-formula img-formula" width="25" height="12" src="https://math.fontein.de/formulae/Lb.TF5g5RL89khbO91d.TLD5QbDvV0.GM3InQA.svgz" alt="\catEns" title="\catEns"></span>), the category of groups by <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="16" src="https://math.fontein.de/formulae/fuj0xeAJAbJYHrp_4.B.un8wkA7ZYPBfJvX47w.svgz" alt="\catGrp" title="\catGrp"></span>, the category of abelian groups by <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/3VO50a7dSjV106cpVTicUgwuuyRbMSvCH7_6Jw.svgz" alt="\catAb" title="\catAb"></span>, the category of rings by <span class="inline-formula"><img class="img-inline-formula img-formula" width="36" height="16" src="https://math.fontein.de/formulae/czkZuiEWfqjHX6VTJ55neZcQD6q1Tn0I7.nZVA.svgz" alt="\catRing" title="\catRing"></span>, the category of field extensions of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="18" src="https://math.fontein.de/formulae/65xTFjisUzxgOyLlFerPiVm0XQrqibnPVNJ8.w.svgz" alt="\catFld(K)" title="\catFld(K)"></span> and, for a ring <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>, the category of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/vYu2wcShMlDKJ.IrmXbb2no6ZOHI_2bIn_7ZWQ.svgz" alt="R" title="R"></span>-modules by <span class="inline-formula"><img class="img-inline-formula img-formula" width="59" height="18" src="https://math.fontein.de/formulae/R1kI8Gy9SNKsUeJoXOU.WvXlPyanbjFb4_o.6Q.svgz" alt="\catMod(R)" title="\catMod(R)"></span>. We begin with the definition of a representable functor. For any category <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> whose objects are sets with further structures, and whose morphisms are maps between these sets, we call <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> <a href="https://en.wikipedia.org/wiki/Concrete_category">concrete</a> and we have the <em>forgetful functor</em> <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="16" src="https://math.fontein.de/formulae/WC5raN6viFoCXcfO0ZkOMJjBoO7QFZHUjhzu2Q.svgz" alt="Forget : \schmC \to \catSet" title="Forget : \schmC \to \catSet"></span>. Finally, denote by <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/ay8BZG4JsCCmPsdXha2Oy3EE.oaLeTmVZ7mIQQ.svgz" alt="\schmC^{op}" title="\schmC^{op}"></span> the <a href="https://en.wikipedia.org/wiki/Opposite_category">opposite</a> category of <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span>; we will not use the term of contravariant functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="12" src="https://math.fontein.de/formulae/98Z6zv2weFMcCClimc1_G.PgJ1o234Ub30sNaw.svgz" alt="\schmC \to \schmD" title="\schmC \to \schmD"></span>, but treat them as functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="12" src="https://math.fontein.de/formulae/0WzkzplV2ffgJ7eQm.oSzDgSK7293k0130UPcA.svgz" alt="\schmC^{op} \to \schmD" title="\schmC^{op} \to \schmD"></span>.
</p>
<p>
Let us begin withe the definition of representable functors.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<ol class="enum-level-1">
<li>
<p>
We say that a functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="103" height="12" src="https://math.fontein.de/formulae/SA3wFbzmMMlj_p79G4hfvh5keTdmr0TWhTj69Q.svgz" alt="F : \schmC^{op} \to \catSet" title="F : \schmC^{op} \to \catSet"></span> is <em>represented</em> by an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span> if there exists a <a href="https://en.wikipedia.org/wiki/Natural_transformation">natural equivalence</a> <span class="inline-formula"><img class="img-inline-formula img-formula" width="164" height="18" src="https://math.fontein.de/formulae/gRG8lA2Rc7dHfJalyVjuP_1R5QbwF6.LJBZArQ.svgz" alt="\eta : F \to \Hom_\schmC(-, X)" title="\eta : F \to \Hom_\schmC(-, X)"></span>.
</p>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="97" height="12" src="https://math.fontein.de/formulae/0XV9z2x4vMXlGcUyNDPpjIDJE1W_KeUVxgdEbQ.svgz" alt="F : \schmC^{op} \to \schmD" title="F : \schmC^{op} \to \schmD"></span> is a functor, <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="12" src="https://math.fontein.de/formulae/L9BylzyUdbJWWZWDypxc51bGXso.8MscmQATCw.svgz" alt="\schmD" title="\schmD"></span> is concrete, we say that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/YONOJBogAF6CR7yxQzTqbuid2nML9DLXhxZHTA.svgz" alt="F" title="F"></span> is represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="174" height="16" src="https://math.fontein.de/formulae/ZPwBlWXxlqV_cTcmcJGI0ZUohKEKfSjoVF1Pqw.svgz" alt="Forget \circ F : \schmC^{op} \to \catSet" title="Forget \circ F : \schmC^{op} \to \catSet"></span> is represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>. Finally, we say that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/YONOJBogAF6CR7yxQzTqbuid2nML9DLXhxZHTA.svgz" alt="F" title="F"></span> is <em>representable</em> if an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span> exists such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/YONOJBogAF6CR7yxQzTqbuid2nML9DLXhxZHTA.svgz" alt="F" title="F"></span> is represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>.
</p>
</li>
<li>
<p>
We say that a functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="88" height="12" src="https://math.fontein.de/formulae/evj89RaycTlHauv9D4ulpfV3t4CFH5i1CgqYNQ.svgz" alt="F : \schmC \to \catSet" title="F : \schmC \to \catSet"></span> is <em>represented</em> by an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span> if there exists a <a href="https://en.wikipedia.org/wiki/Natural_transformation">natural equivalence</a> <span class="inline-formula"><img class="img-inline-formula img-formula" width="163" height="18" src="https://math.fontein.de/formulae/dXKh_zB1aa4c34cvYmJgYz0hDss6ELuTi5g3uQ.svgz" alt="\eta : F \to \Hom_\schmC(X, -)" title="\eta : F \to \Hom_\schmC(X, -)"></span>.
</p>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="12" src="https://math.fontein.de/formulae/3VdmMLQRDdvaBF6wezJzj4oRJ6tJBlDLLtlfMw.svgz" alt="F : \schmC \to \schmD" title="F : \schmC \to \schmD"></span> is a functor, <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="12" src="https://math.fontein.de/formulae/L9BylzyUdbJWWZWDypxc51bGXso.8MscmQATCw.svgz" alt="\schmD" title="\schmD"></span> is concrete, we say that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/YONOJBogAF6CR7yxQzTqbuid2nML9DLXhxZHTA.svgz" alt="F" title="F"></span> is represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="159" height="16" src="https://math.fontein.de/formulae/WxjS.XVckUvzlw4qFNFlrQS.LQQ3PbOuEdMlOQ.svgz" alt="Forget \circ F : \schmC \to \catSet" title="Forget \circ F : \schmC \to \catSet"></span> is represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>. Finally, we say that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/YONOJBogAF6CR7yxQzTqbuid2nML9DLXhxZHTA.svgz" alt="F" title="F"></span> is <em>representable</em> if an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span> exists such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/YONOJBogAF6CR7yxQzTqbuid2nML9DLXhxZHTA.svgz" alt="F" title="F"></span> is represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>.
</p>
</li>
</ol>
</div>
</div>
<p>
This sounds rather abstract. Let us describe <span class="inline-formula"><img class="img-inline-formula img-formula" width="187" height="18" src="https://math.fontein.de/formulae/K3s4OuKRqrM_3tcTbQvp6HZKwFd.5JrLYUJ6Vw.svgz" alt="\Hom_\schmC(-, X) : \schmC^{op} \to \catSet" title="\Hom_\schmC(-, X) : \schmC^{op} \to \catSet"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="171" height="18" src="https://math.fontein.de/formulae/tcVjF45FhSVmsiGU8ZCDYuSWW_j_DBVfQNjRmA.svgz" alt="\Hom_\schmC(X, -) : \schmC \to \catSet" title="\Hom_\schmC(X, -) : \schmC \to \catSet"></span> in a little more in detail. To an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/9tpiazplIU4lT9qQJZQvtmT.jkfAQYbAsv_qfQ.svgz" alt="A \in \schmC" title="A \in \schmC"></span>, they assign the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="18" src="https://math.fontein.de/formulae/qhiblmE72ujpT0vqdCGrhPCI1qJ76F4Euo4eqw.svgz" alt="\Hom_\schmC(A, X)" title="\Hom_\schmC(A, X)"></span> of all morphisms from <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> to <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>, respectively the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="97" height="18" src="https://math.fontein.de/formulae/WvY5C.IvFeBaezxzvxM971OFXSYGyypGFwYzQg.svgz" alt="\Hom_\schmC(X, A)" title="\Hom_\schmC(X, A)"></span> of all morphisms from <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> to <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span>. And to a morphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="17" src="https://math.fontein.de/formulae/Y.AClLlF2mv0FNOKF1f6k7Thq3uaI93Q032lDQ.svgz" alt="\varphi : A \to A'" title="\varphi : A \to A'"></span> they assign the map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="445" height="19" src="https://math.fontein.de/formulae/zODXm8MFS7QyHjsoRZZc_kLh9tGw.bxbiGAi4w.svgz" alt="\Hom_\schmC(\varphi, X) : \Hom_\schmC(A', X) \to \Hom_\schmC(A, X), \quad f \mapsto f \circ \varphi" title="\Hom_\schmC(\varphi, X) : \Hom_\schmC(A', X) \to \Hom_\schmC(A, X), \quad f \mapsto f \circ \varphi">
</div>
<p>
respectively
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="446" height="19" src="https://math.fontein.de/formulae/SUJyvuR3kw.BU54mkTII2Yi9DjSaILWFnVrLcQ.svgz" alt="\Hom_\schmC(X, \varphi) : \Hom_\schmC(X, A) \to \Hom_\schmC(X, A'), \quad f \mapsto \varphi \circ f." title="\Hom_\schmC(X, \varphi) : \Hom_\schmC(X, A) \to \Hom_\schmC(X, A'), \quad f \mapsto \varphi \circ f.">
</div>
<p>
We want to make both concepts more clear with two important examples.
</p>
<div class="theorem-environment theorem-example-environment">
<div class="theorem-header theorem-example-header">
Example.
</div>
<div class="theorem-content theorem-example-content">
<p>
Let us consider the category <span class="inline-formula"><img class="img-inline-formula img-formula" width="101" height="18" src="https://math.fontein.de/formulae/Bh2yd9vxlekfQ9MFGpLqpNY_p2JF4CXFztUyqA.svgz" alt="\schmC = \catRing(K)" title="\schmC = \catRing(K)"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-algebras, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> is a fixed ring (or field, if you feel more comfortable). Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="12" src="https://math.fontein.de/formulae/haQiBBQTBb87jlSrgLzeHvraw7OmAtyEA3iHCA.svgz" alt="X = K" title="X = K"></span> be the ring itself, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="140" height="18" src="https://math.fontein.de/formulae/dMI0CgDEmMBnO2cr691P3iM0B9gZv9Ms.XWR_g.svgz" alt="A = K[x_1, \dots, x_n]" title="A = K[x_1, \dots, x_n]"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="246" height="18" src="https://math.fontein.de/formulae/mqcWyLGwNIYWrVgdkgeF5Fn2EURxSqkUlPMQfQ.svgz" alt="A' = K[x_1, \dots, x_n]/(f_1, \dots, f_m)" title="A' = K[x_1, \dots, x_n]/(f_1, \dots, f_m)"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="203" height="18" src="https://math.fontein.de/formulae/YpkCGIhgZrDwLigt_OlF8zi8K7nOTPkaU1jTOQ.svgz" alt="f_1, \dots, f_m \in K[x_1, \dots, x_n]" title="f_1, \dots, f_m \in K[x_1, \dots, x_n]"></span> are polynomials. Now every <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="16" src="https://math.fontein.de/formulae/wRMp5KxF_Rx4lBHN1.7B.uEsP4MRG8UWINrtbw.svgz" alt="\varphi : A \to K" title="\varphi : A \to K"></span> is a substitution homomorphism, i.e. there exists a uniquely determined <span class="inline-formula"><img class="img-inline-formula img-formula" width="170" height="18" src="https://math.fontein.de/formulae/aWQ4C8ZJW0x36_KQJclIsvN93B9O2KxnYx3iHg.svgz" alt="a = (a_1, \dots, a_n) \in K^n" title="a = (a_1, \dots, a_n) \in K^n"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="160" height="18" src="https://math.fontein.de/formulae/j9BBboTG5h0swdOZ.BY5Xa6ialrUhIpe9VvzzA.svgz" alt="\varphi(f) = f(a_1, \dots, a_n)" title="\varphi(f) = f(a_1, \dots, a_n)"></span>. Conversely, for any tuple <span class="inline-formula"><img class="img-inline-formula img-formula" width="170" height="18" src="https://math.fontein.de/formulae/aWQ4C8ZJW0x36_KQJclIsvN93B9O2KxnYx3iHg.svgz" alt="a = (a_1, \dots, a_n) \in K^n" title="a = (a_1, \dots, a_n) \in K^n"></span>, there exists a homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="16" src="https://math.fontein.de/formulae/kvQhTXOOAkKXM.4bnelgHADcgwZcRCsmIXb7Sg.svgz" alt="\varphi_a : A \to K" title="\varphi_a : A \to K"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="18" src="https://math.fontein.de/formulae/hQj80Hqf1xmOX4CBeWYo5mK4vDRaEm.FL9EI_w.svgz" alt="f \mapsto f(a)" title="f \mapsto f(a)"></span>. Hence, we can identify <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="18" src="https://math.fontein.de/formulae/1ov6.chEMyPkbIs_v7cCKQJuhURbIPaHoJDs3g.svgz" alt="\Hom_\schmC(A, K)" title="\Hom_\schmC(A, K)"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="26" height="12" src="https://math.fontein.de/formulae/omSZXy11nxsn8Xbfa39ocK8yTAKexIBDz3e8yw.svgz" alt="K^n" title="K^n"></span>.
</p>
<p>
Now the homomorphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="117" height="18" src="https://math.fontein.de/formulae/N0LEsnVQQyN.U8FZft53rypJskFgdGoVIKyr0A.svgz" alt="A' = A/I \to K" title="A' = A/I \to K"></span> correspond to the homomorphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="12" src="https://math.fontein.de/formulae/UiV3TkXNe7RWAAvmlrr73ZmKdOYoDsy12N.RDw.svgz" alt="A \to K" title="A \to K"></span> whose kernel contain <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="12" src="https://math.fontein.de/formulae/f15cBiWrTTbfiyc5LOlcnCn.Yan56zeX76_Hlg.svgz" alt="I" title="I"></span>. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="16" src="https://math.fontein.de/formulae/kvQhTXOOAkKXM.4bnelgHADcgwZcRCsmIXb7Sg.svgz" alt="\varphi_a : A \to K" title="\varphi_a : A \to K"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="72" height="18" src="https://math.fontein.de/formulae/hQj80Hqf1xmOX4CBeWYo5mK4vDRaEm.FL9EI_w.svgz" alt="f \mapsto f(a)" title="f \mapsto f(a)"></span> be a homomorphism. Then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="378" height="18" src="https://math.fontein.de/formulae/iiLRtwXMMF00hStymSWoZ_nePMAAmK0YuabdNA.svgz" alt="I \subseteq \ker \varphi_a \Leftrightarrow \forall i : f_i(a) = 0 \Leftrightarrow a \in V(f_1, \dots, f_m)," title="I \subseteq \ker \varphi_a \Leftrightarrow \forall i : f_i(a) = 0 \Leftrightarrow a \in V(f_1, \dots, f_m),">
</div>
<p>
where
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="452" height="18" src="https://math.fontein.de/formulae/Sr1cNfKkuNVom07KjuuBAOhFQIdvKKezie0nuw.svgz" alt="V := V(f_1, \dots, f_m) := \{ a \in K^n \mid f_1(a) = \dots = f_m(a) = 0 \}" title="V := V(f_1, \dots, f_m) := \{ a \in K^n \mid f_1(a) = \dots = f_m(a) = 0 \}">
</div>
<p>
is the <a href="https://en.wikipedia.org/wiki/Algebraic_variety">variety</a> defined by the polynonials <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="16" src="https://math.fontein.de/formulae/v6FL0B4yaEdbdtZLD8G_fQp4MpnY3_HZUYy9zw.svgz" alt="f_1, \dots, f_m" title="f_1, \dots, f_m"></span>. Hence, we can identify <span class="inline-formula"><img class="img-inline-formula img-formula" width="103" height="18" src="https://math.fontein.de/formulae/yZZrj_ZvJS0xWoR2YrqTrGXCjxYC1dgWr.fyyg.svgz" alt="\Hom_\schmC(A', K)" title="\Hom_\schmC(A', K)"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/_NfRTOgxDFggbO2SNYKEhtwhL_2qpB0USZxZUA.svgz" alt="V" title="V"></span>. Now consider the projection <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="13" src="https://math.fontein.de/formulae/e1Ywpq6rWLayBedeuypx4PPX1littUpSgMqAaw.svgz" alt="\pi : A \to A'" title="\pi : A \to A'"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="80" height="16" src="https://math.fontein.de/formulae/O_U3gLfxcLxyY6FVNil0NwUzdPFU0.52xqkL1Q.svgz" alt="f \mapsto f + I" title="f \mapsto f + I"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="136" height="18" src="https://math.fontein.de/formulae/5sGghJDi9NIpAxiOseIi5US_d4fEbtk6MLZzHg.svgz" alt="\varphi \in \Hom_\schmC(A', K)" title="\varphi \in \Hom_\schmC(A', K)"></span> corresponds to the point <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="13" src="https://math.fontein.de/formulae/QvCYRy_033Ts28yemsbsPEtD8Hnn_pnYPLz8Lw.svgz" alt="a \in V" title="a \in V"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="256" height="18" src="https://math.fontein.de/formulae/U28DyvzqGMCs..BS8QU9vnLyBtpgVh7NSKa1sg.svgz" alt="\Hom_\schmC(\pi, K)(\varphi) = \varphi \circ \pi : A \to K" title="\Hom_\schmC(\pi, K)(\varphi) = \varphi \circ \pi : A \to K"></span> also corresponds to <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="8" src="https://math.fontein.de/formulae/azRcBM3xT5fNJHpsWOTZlxYAFAmaA0iJVK1nog.svgz" alt="a" title="a"></span>, but this time <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="13" src="https://math.fontein.de/formulae/Yz8kzFyI.J7fCm_6jpQNmQOTr1.fMfmXqrrM8g.svgz" alt="a \in K^n" title="a \in K^n"></span>! Hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="18" src="https://math.fontein.de/formulae/b3fE6lohn4G9QBgaTIErrzhazlmtrW8d.Iw.3g.svgz" alt="\Hom_\schmC(\pi, K)(\varphi)" title="\Hom_\schmC(\pi, K)(\varphi)"></span> is the inclusion map <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="12" src="https://math.fontein.de/formulae/lXcX9OS5Ve8PtolHjm6IXyD1RPxD3t23HJGHvQ.svgz" alt="V \injto K^n" title="V \injto K^n"></span>.
</p>
<p>
Now, if one replaces <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="18" src="https://math.fontein.de/formulae/PocTPdXovQIi9U9d3JLEEbFIFoyj.tCepoSnhg.svgz" alt="\Hom_\schmC(-, K)" title="\Hom_\schmC(-, K)"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="18" src="https://math.fontein.de/formulae/DzpHMR4OiFsjEjjg06LmSxSwYZfzJV1YRgKNgw.svgz" alt="\Hom_\schmC(-, S)" title="\Hom_\schmC(-, S)"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> is a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-algebra, we can identify <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="18" src="https://math.fontein.de/formulae/UiBhYy848DulfrRRvarVimTi.HS43Ahb0ZdSvw.svgz" alt="\Hom_\schmC(A, S)" title="\Hom_\schmC(A, S)"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="12" src="https://math.fontein.de/formulae/5Rpv3Mm5UgfjFMN.cxG0gx_01JemY61bNXAMzA.svgz" alt="S^n" title="S^n"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="18" src="https://math.fontein.de/formulae/gOUW9pGjHNV.HRyWN7IlyWakOoOuVmmNeqQjeA.svgz" alt="\Hom_\schmC(A', S)" title="\Hom_\schmC(A', S)"></span> with
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="465" height="18" src="https://math.fontein.de/formulae/AdLD3dKl.qloANrWzGC4E_pbl466QRLVVsvb1w.svgz" alt="V_S := V_S(f_1, \dots, f_m) := \{ a \in S^n \mid f_1(a) = \dots = f_m(a) = 0 \}," title="V_S := V_S(f_1, \dots, f_m) := \{ a \in S^n \mid f_1(a) = \dots = f_m(a) = 0 \},">
</div>
<p>
and again <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="18" src="https://math.fontein.de/formulae/sXq52nKWKo9O9OfZoBHDmqjXDcMno0Qmf3cI1w.svgz" alt="\Hom_\schmC(\pi, S)" title="\Hom_\schmC(\pi, S)"></span> is the inclusion map <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="15" src="https://math.fontein.de/formulae/p5EAKu4sfMGEKOcUG_MKyu5ejkbASpf8TeTl6Q.svgz" alt="V_S \injto S^n" title="V_S \injto S^n"></span>. Note that this is not just a toy example, but a very fundamental concept used with <a href="https://en.wikipedia.org/wiki/Affine_scheme">affine schemes</a>. In fact, there, one fixes a <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-algebra <span class="inline-formula"><img class="img-inline-formula img-formula" width="18" height="13" src="https://math.fontein.de/formulae/RHaEFiJ80k7jYc_emQtM.QyRfF7GZDZ7Wc4RTA.svgz" alt="A'" title="A'"></span>, say our <span class="inline-formula"><img class="img-inline-formula img-formula" width="246" height="18" src="https://math.fontein.de/formulae/mqcWyLGwNIYWrVgdkgeF5Fn2EURxSqkUlPMQfQ.svgz" alt="A' = K[x_1, \dots, x_n]/(f_1, \dots, f_m)" title="A' = K[x_1, \dots, x_n]/(f_1, \dots, f_m)"></span>, and looks at the functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="174" height="18" src="https://math.fontein.de/formulae/jdfP6XINdgsbzlUb53xrvG3SML1Hx6hXQnjp9g.svgz" alt="\Hom_\schmC(A', -) : \schmC \to \catSet" title="\Hom_\schmC(A', -) : \schmC \to \catSet"></span>; this functor is called the <a href="https://en.wikipedia.org/wiki/Functor_of_points">functor of points</a> as it assigns to every <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-algebra <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> (indirectly) the solutions <span class="inline-formula"><img class="img-inline-formula img-formula" width="112" height="18" src="https://math.fontein.de/formulae/U5UTegVPdIX0foVEk8UjE6CvAB1xH2BFqehlMw.svgz" alt="V_S(f_1, \dots, f_m)" title="V_S(f_1, \dots, f_m)"></span> of the polynomial equations <span class="inline-formula"><img class="img-inline-formula img-formula" width="185" height="18" src="https://math.fontein.de/formulae/3MuilHv2NiZTWwVsZ_OjKKiS9xjB4F06Qh4T9A.svgz" alt="f_1(a) = \dots = f_m(a) = 0" title="f_1(a) = \dots = f_m(a) = 0"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="53" height="13" src="https://math.fontein.de/formulae/TUUrOx8x8uErj.EXdSde8.7CYBMPvlA7o8HwwQ.svgz" alt="a \in S^n" title="a \in S^n"></span>. In fact, if we consider the functor
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="270" height="18" src="https://math.fontein.de/formulae/hi6wH55.lSkzLE8F6XKtDBTUpFPOS4mterbYkA.svgz" alt="V_\bullet : \schmC \to \catSet, \quad S \mapsto V_S(f_1, \dots, f_m)" title="V_\bullet : \schmC \to \catSet, \quad S \mapsto V_S(f_1, \dots, f_m)">
</div>
<p>
with <span class="inline-formula"><img class="img-inline-formula img-formula" width="108" height="17" src="https://math.fontein.de/formulae/nzZ7tp27zxJKVRHwMFVU4GA7ybOYkWtJAme8GA.svgz" alt="V_\phi : V_S \to V_{S'}" title="V_\phi : V_S \to V_{S'}"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="255" height="18" src="https://math.fontein.de/formulae/pL5JvngDim1R8x.XhOZNzM6UaEJ03yV7lEyVBA.svgz" alt="(a_1, \dots, a_n) \mapsto (\phi(a_1), \dots, \phi(a_n))" title="(a_1, \dots, a_n) \mapsto (\phi(a_1), \dots, \phi(a_n))"></span> for <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="17" src="https://math.fontein.de/formulae/stg74DBzV_3m6YR_Qo9g8a3.fO2moBxWedmEdw.svgz" alt="\phi : S \to S'" title="\phi : S \to S'"></span>, then we obtain a natural transformation
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="166" height="19" src="https://math.fontein.de/formulae/IVO6gmgInQeHwK4buyDT2KDLpeNlLYkoNWPTVA.svgz" alt="\eta : V_\bullet \to \Hom_\schmC(A', \bullet)" title="\eta : V_\bullet \to \Hom_\schmC(A', \bullet)">
</div>
<p>
which assigns to <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="13" src="https://math.fontein.de/formulae/iabjkHSA3OvgH10Gut5YegEG7I4sx6OmYiQjEQ.svgz" alt="S \in \schmC" title="S \in \schmC"></span> the map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="299" height="32" src="https://math.fontein.de/formulae/a2FjpHfbvs5D5S79v909qsQ1VERtR2GaFblp8g.svgz" alt="V_S \to \Hom_\schmC(A', S), \quad a \mapsto \begin{cases} A' \to S \atop f \mapsto f(a). \end{cases}" title="V_S \to \Hom_\schmC(A', S), \quad a \mapsto \begin{cases} A' \to S \atop f \mapsto f(a). \end{cases}">
</div>
<p>
Since by the above, this is a bijection, it turns out that <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/LJX59haeTalq4bSnxWApTe0lMbaooEzjD2TZqQ.svgz" alt="\eta" title="\eta"></span> is in fact a <em>natural equivalence</em>; hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="18" height="13" src="https://math.fontein.de/formulae/RHaEFiJ80k7jYc_emQtM.QyRfF7GZDZ7Wc4RTA.svgz" alt="A'" title="A'"></span> represents the functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="18" height="15" src="https://math.fontein.de/formulae/4oc8LHyt1aszdEIUQqlCVR1czpxcQFBNpJGmBg.svgz" alt="V_\bullet" title="V_\bullet"></span>.
</p>
</div>
</div>
<div class="theorem-environment theorem-example-environment">
<div class="theorem-header theorem-example-header">
Example.
</div>
<div class="theorem-content theorem-example-content">
<p>
Another example also comes from algebraic geometry, namely <a href="https://en.wikipedia.org/wiki/Elliptic_curve">elliptic curves</a>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/yFrTB12ITTLe1DP3TnmeXVBU1JI15EuP_XUWsQ.svgz" alt="E : y^2 = x^3 + a x + b" title="E : y^2 = x^3 + a x + b"></span> is an elliptic curve defined over a field <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="16" src="https://math.fontein.de/formulae/heQcBVUHsACyFDRsIA0nqyKgu4TMUYO62R2EdQ.svgz" alt="a, b \in K" title="a, b \in K"></span>, then for every field extension <span class="inline-formula"><img class="img-inline-formula img-formula" width="37" height="18" src="https://math.fontein.de/formulae/2X12W49lj02TDhOLBRCb3NHtYkwNtYH0Fovx6A.svgz" alt="L/K" title="L/K"></span> we can define
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="331" height="20" src="https://math.fontein.de/formulae/H8mcpDVd99bA5hnhf7aNJ27L.JCshhwHqNdzxQ.svgz" alt="E(L) := \{ (x, y) \mid y^2 = x^3 + a x + b \} \cup \{ \infty \}." title="E(L) := \{ (x, y) \mid y^2 = x^3 + a x + b \} \cup \{ \infty \}.">
</div>
<p>
For every <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span>-homomorphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="82" height="17" src="https://math.fontein.de/formulae/IGxlPNR9Ho3Q82eUgpHkgagAflk4qT0aecD3aQ.svgz" alt="\phi : L \to L'" title="\phi : L \to L'"></span>, we obtain a map <span class="inline-formula"><img class="img-inline-formula img-formula" width="166" height="18" src="https://math.fontein.de/formulae/UkqLi6ofD4uy8m7OMFkv3HR6ay43MyVzOcc1zw.svgz" alt="E(\phi) : E(L) \to E(L')" title="E(\phi) : E(L) \to E(L')"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="158" height="18" src="https://math.fontein.de/formulae/Sk4t4tk0aJFkDMg83aKGx6ISN9O67mrFzmQqbA.svgz" alt="(x, y) \mapsto (\phi(x), \phi(y))" title="(x, y) \mapsto (\phi(x), \phi(y))"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="8" src="https://math.fontein.de/formulae/lA.62FPvJ2KliixCWPzjUGsNNu5LiQK6cAu.4Q.svgz" alt="\infty \mapsto \infty" title="\infty \mapsto \infty"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="144" height="18" src="https://math.fontein.de/formulae/8rdOuIidu2KbFRNpVi2sB3he8JaYBOGci_OWeA.svgz" alt="E : \catFld(K) \to \catAb" title="E : \catFld(K) \to \catAb"></span> is a functor into the category of abelian groups!
</p>
<p>
Now one can ask whether the functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/nxZcONSD_fgUzAIJDOzRaD8li6np2krjuBURhw.svgz" alt="E" title="E"></span> is representable, i.e. whether there exists a field extension <span class="inline-formula"><img class="img-inline-formula img-formula" width="103" height="18" src="https://math.fontein.de/formulae/2CgDBJet1Gq2OqOJC_Kytb44cAeA9dVDNkczbg.svgz" alt="L \in \catFld(K)" title="L \in \catFld(K)"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/08bW5Zvy2ST6Ewwt6yOyAbfn7ZY0nrbV5GNE.Q.svgz" alt="K" title="K"></span> such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="145" height="20" src="https://math.fontein.de/formulae/FEUIvezsdQj8PkOur81phSQouwpiHJatH_U3cA.svgz" alt="\Hom_{\catFld(K)}(L', L)" title="\Hom_{\catFld(K)}(L', L)"></span> can be <em>identified in a natural way</em> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="18" src="https://math.fontein.de/formulae/LQDT007XsfHJZwx6PhQEvnUq8OcbymsYBDfGnA.svgz" alt="E(L')" title="E(L')"></span> for every field extension <span class="inline-formula"><img class="img-inline-formula img-formula" width="42" height="18" src="https://math.fontein.de/formulae/iE4dHNrRfG6qt5MkVReyYgIN0V9CqWF7oGnPmw.svgz" alt="L'/K" title="L'/K"></span> – that is exactly what it means for <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/YONOJBogAF6CR7yxQzTqbuid2nML9DLXhxZHTA.svgz" alt="F" title="F"></span> being represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="17" height="13" src="https://math.fontein.de/formulae/5bbGVwv62EiGXc1VwhpjEJ0ccJnJZ2GTCXFxFg.svgz" alt="L'" title="L'"></span>. Unfortunately, it turns out not to be possible.
</p>
</div>
</div>
<p>
For two categories <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="15" height="12" src="https://math.fontein.de/formulae/L9BylzyUdbJWWZWDypxc51bGXso.8MscmQATCw.svgz" alt="\schmD" title="\schmD"></span>, we can consider the category <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="18" src="https://math.fontein.de/formulae/UuYmbkFwZfk02tPNZac.qXZ2CP5lTk3iWeyo2w.svgz" alt="\Hom(\schmC^{op}, \schmD)" title="\Hom(\schmC^{op}, \schmD)"></span>, whose objects are functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="68" height="12" src="https://math.fontein.de/formulae/0WzkzplV2ffgJ7eQm.oSzDgSK7293k0130UPcA.svgz" alt="\schmC^{op} \to \schmD" title="\schmC^{op} \to \schmD"></span> and whose morphisms are natural transformations of such functors.
</p>
<p>
If we fix a category <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> and an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span>, we obtain the functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="151" height="18" src="https://math.fontein.de/formulae/6Pl3cGIlRBv6tPPpgeao0OoNxMY96mspTMYPTw.svgz" alt="h_X := \Hom_\schmC(-, X)" title="h_X := \Hom_\schmC(-, X)"></span>, which is an element of <span class="inline-formula"><img class="img-inline-formula img-formula" width="105" height="18" src="https://math.fontein.de/formulae/EQ6NmR3q5f0dfCIpetMoKF2mjVjBj2tRIfaDHw.svgz" alt="\Hom(\schmC^{op}, \catSet)" title="\Hom(\schmC^{op}, \catSet)"></span>. If we have another object <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/.RRmrfaMyBlSwvNNuQvc0Y16VVj6rTIqIJCqZw.svgz" alt="Y" title="Y"></span> together with a morphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="16" src="https://math.fontein.de/formulae/fKJCafTOPj9M6sUSIkbL0_HivXhzL9Yqk777Mw.svgz" alt="f : X \to Y" title="f : X \to Y"></span>, we obtain a natural transformation <span class="inline-formula"><img class="img-inline-formula img-formula" width="108" height="17" src="https://math.fontein.de/formulae/NqwJ9H1STdyayy141fKsdIRw1pugQWCgh81nRw.svgz" alt="h_f : h_X \to h_Y" title="h_f : h_X \to h_Y"></span> by assigning <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/zA2fDCK.wSX1JlPEllSPBENcV2tamJb36gtWIg.svgz" alt="Z \in \schmC" title="Z \in \schmC"></span> the morphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="220" height="18" src="https://math.fontein.de/formulae/Pl898n5ntmfvEKesjnfP7Iqljzs2gaOUOhJivA.svgz" alt="\Hom_\schmC(Z, X) \to \Hom_\schmC(Z, Y)" title="\Hom_\schmC(Z, X) \to \Hom_\schmC(Z, Y)"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="73" height="16" src="https://math.fontein.de/formulae/m2VYrsV_9ArR4Vp5QmAFqOXI9X4Fa9vc03HZ2A.svgz" alt="g \mapsto f \circ g" title="g \mapsto f \circ g"></span>. Hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span> can be seen as a functor from <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> to <span class="inline-formula"><img class="img-inline-formula img-formula" width="144" height="22" src="https://math.fontein.de/formulae/VLx1KF0iYMnX1apUAdfvb.YqC0HQXJldbEfpFg.svgz" alt="\Hom(\schmC^{op}, \catSet) =: \hat{\schmC}" title="\Hom(\schmC^{op}, \catSet) =: \hat{\schmC}"></span>. This functor is also called the <em>(contravariant) Yoneda embedding</em>.
</p>
<div class="theorem-environment theorem-theorem-environment">
<div class="theorem-header theorem-theorem-header">
<a name="yoneda-lemma"></a>
Theorem (Yoneda's Lemma).
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> be a category and <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="18" src="https://math.fontein.de/formulae/HY2.U1y7o.VaLzfw14uMyV1TzpRpbm55.fMqww.svgz" alt="H \in \hat{\schmC}" title="H \in \hat{\schmC}"></span>. Then there is a natural bijection
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="182" height="19" src="https://math.fontein.de/formulae/AU4B6iesGFW78ejijpcTn9xLvZGrLEv6lfSd3Q.svgz" alt="H(X) \to \Hom_{\hat{\schmC}}(h_X, H)" title="H(X) \to \Hom_{\hat{\schmC}}(h_X, H)">
</div>
<p>
with the following property:
</p>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="18" src="https://math.fontein.de/formulae/s6FePyyeCb5DV497Xa7ypI6GaAAbgSDxQJRldw.svgz" alt="u \in H(X)" title="u \in H(X)"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="8" src="https://math.fontein.de/formulae/Ei1EDDiFBqhkFoiwvRad2hcmf_KVutGWDniJpA.svgz" alt="u" title="u"></span> is mapped onto the natural transformation <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="15" src="https://math.fontein.de/formulae/a0Nn6COzd9zKHk6tkRx817xLpEj4ueUh2jIZJQ.svgz" alt="h_X \to H" title="h_X \to H"></span>, which satisfies that <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/mEXTAecG9bdu5.f72xqv9l5iXmDrva2j0orOBQ.svgz" alt="g \in h_X(A)" title="g \in h_X(A)"></span> is mapped onto <span class="inline-formula"><img class="img-inline-formula img-formula" width="128" height="18" src="https://math.fontein.de/formulae/AURy21NMoFhmZYwVkDKSW7M5ecBQAE0evCuykA.svgz" alt="H(g)(u) \in H(A)" title="H(g)(u) \in H(A)"></span>.
</p>
<p>
In other terms:
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="350" height="131" src="https://math.fontein.de/formulae/asU1VX_SBX87jKsm3T3DvmLp9BGArKJQHP6C_Q.svgz" alt="\eta :{} & \bullet_2(\bullet_1) \to \Hom_{\hat{\schmC}}(h(\bullet_1), \bullet_2), \\
& (X, H) \mapsto \begin{cases} H(X) \to \Hom_{\hat{\schmC}}(h_X, H) \\ u \mapsto \begin{cases} h_X \to H \\ A \mapsto \begin{cases} h_X(A) \to H(A) \\ g \mapsto H(g)(u). \end{cases} \end{cases} \end{cases}" title="\eta :{} & \bullet_2(\bullet_1) \to \Hom_{\hat{\schmC}}(h(\bullet_1), \bullet_2), \\
& (X, H) \mapsto \begin{cases} H(X) \to \Hom_{\hat{\schmC}}(h_X, H) \\ u \mapsto \begin{cases} h_X \to H \\ A \mapsto \begin{cases} h_X(A) \to H(A) \\ g \mapsto H(g)(u). \end{cases} \end{cases} \end{cases}">
</div>
<p>
is a natural equivalence.
</p>
</div>
</div>
<p>
This looks rather complicated. A simpler corollary is:
</p>
<div class="theorem-environment theorem-corollary-environment">
<div class="theorem-header theorem-corollary-header">
<a name="yoneda-corollary"></a>
Corollary.
</div>
<div class="theorem-content theorem-corollary-content">
<p>
The Yoneda embedding <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="17" src="https://math.fontein.de/formulae/3z1aFiDAKL2wSl_z1pk7IBMWfaDw4QlXtdNJgg.svgz" alt="h : \schmC \to \hat{\schmC}" title="h : \schmC \to \hat{\schmC}"></span> is <a href="https://en.wikipedia.org/wiki/Faithful_functor">fully faithful</a>, i.e. for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="16" src="https://math.fontein.de/formulae/nn_Qx0YGNdNQ.McKxQQe0g.e4iyvVCkR.LS4Ng.svgz" alt="X, Y \in \schmC" title="X, Y \in \schmC"></span>, the map <span class="inline-formula"><img class="img-inline-formula img-formula" width="265" height="19" src="https://math.fontein.de/formulae/KKoumkTafaIlgyvlwsi24opj2yXIzPMvTHHQhQ.svgz" alt="h : \Hom_\schmC(X, Y) \to \Hom_{\hat{\schmC}}(h_X, h_Y)" title="h : \Hom_\schmC(X, Y) \to \Hom_{\hat{\schmC}}(h_X, h_Y)"></span> is bijective. In fact, <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span> is a <a href="https://en.wikipedia.org/wiki/Full_subcategory#Embeddings">full embedding</a>.
</p>
</div>
</div>
<p>
Before we continue with the proof of Yoneda's lemma and the corollary, let us first take apart the notation from Yoneda's lemma. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="18" src="https://math.fontein.de/formulae/HY2.U1y7o.VaLzfw14uMyV1TzpRpbm55.fMqww.svgz" alt="H \in \hat{\schmC}" title="H \in \hat{\schmC}"></span>, it means that <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/592A9cpJqEapTTR0mQzYD2FBigGo_84WF1hl8w.svgz" alt="H" title="H"></span> is a functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="12" src="https://math.fontein.de/formulae/bt59FrZWLAmoBo23eX_WujC.WHTcaTd4Qpk3dg.svgz" alt="\schmC^{op} \to \catSet" title="\schmC^{op} \to \catSet"></span>. The class (which, by Yoneda's lemma, is a set) <span class="inline-formula"><img class="img-inline-formula img-formula" width="108" height="19" src="https://math.fontein.de/formulae/uoQuRVNhOhazxAqu6DbybQqrAiOO.1RtERA63w.svgz" alt="\Hom_{\hat{\schmC}}(h_X, H)" title="\Hom_{\hat{\schmC}}(h_X, H)"></span> is the class of natural transformations <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="15" src="https://math.fontein.de/formulae/pHjwPFPvd_7PAAUBN6t5FdpXcMjm86wt3jCAiQ.svgz" alt="h_X \to X" title="h_X \to X"></span>. Now, Yoneda's lemma basically says that there is a natural bijection between the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/cYcHOuFeZnxiQ_M6_eXlkcKqo2CzbRCWPSJPWQ.svgz" alt="H(X)" title="H(X)"></span> and the set of natural transformations <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="15" src="https://math.fontein.de/formulae/a0Nn6COzd9zKHk6tkRx817xLpEj4ueUh2jIZJQ.svgz" alt="h_X \to H" title="h_X \to H"></span>.
</p>
<p>
The property says that the bijection has to look as follows: to an element <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="18" src="https://math.fontein.de/formulae/s6FePyyeCb5DV497Xa7ypI6GaAAbgSDxQJRldw.svgz" alt="u \in H(X)" title="u \in H(X)"></span>, we want to assign a natural transformation <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="16" src="https://math.fontein.de/formulae/25QQTB.yi4YCycjlHrH7i66NYD59CUwAlY8fqA.svgz" alt="\eta_u : h_X \to H" title="\eta_u : h_X \to H"></span> which is defined as follows: for an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/9tpiazplIU4lT9qQJZQvtmT.jkfAQYbAsv_qfQ.svgz" alt="A \in \schmC" title="A \in \schmC"></span>, we want that <span class="inline-formula"><img class="img-inline-formula img-formula" width="181" height="18" src="https://math.fontein.de/formulae/MjW8rUxKNSZB8J.yREADclOAssxM5vK3KeoVJA.svgz" alt="\eta_u(A) : h_X(A) \to H(A)" title="\eta_u(A) : h_X(A) \to H(A)"></span> is defined by <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="18" src="https://math.fontein.de/formulae/7VWXfaNHcyUtSZkGdb.nMMpqwTKlbJeS_F_0oQ.svgz" alt="g \mapsto H(g)(u)" title="g \mapsto H(g)(u)"></span>, i.e. we want to map <span class="inline-formula"><img class="img-inline-formula img-formula" width="129" height="18" src="https://math.fontein.de/formulae/9K2B1pe5n7NRGzmh2srIS1ReHJWsl4CKGL8pJg.svgz" alt="g \in \Hom_\schmC(A, X)" title="g \in \Hom_\schmC(A, X)"></span> to <span class="inline-formula"><img class="img-inline-formula img-formula" width="171" height="18" src="https://math.fontein.de/formulae/.52OrtCHFF1AYk3iI3BW3WXXLhueYOWxmMWbTA.svgz" alt="H(g) : H(X) \to H(A)" title="H(g) : H(X) \to H(A)"></span> evaluated at <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="8" src="https://math.fontein.de/formulae/Ei1EDDiFBqhkFoiwvRad2hcmf_KVutGWDniJpA.svgz" alt="u" title="u"></span>, which was an element of <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="18" src="https://math.fontein.de/formulae/cYcHOuFeZnxiQ_M6_eXlkcKqo2CzbRCWPSJPWQ.svgz" alt="H(X)" title="H(X)"></span>.
</p>
<p>
Still sounds really complicated, doesn't it? Well, lets start with the proof, which includes some fancy diagrams.
</p>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof (of <a href="https://math.fontein.de/2009/08/16/fun-with-representable-functors-or-why-i-like-yondeas-lemma/#yoneda-lemma">Yoneda's Lemma</a>).
</div>
<div class="theorem-content theorem-proof-content">
<p>
First, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="18" src="https://math.fontein.de/formulae/s6FePyyeCb5DV497Xa7ypI6GaAAbgSDxQJRldw.svgz" alt="u \in H(X)" title="u \in H(X)"></span>. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/9tpiazplIU4lT9qQJZQvtmT.jkfAQYbAsv_qfQ.svgz" alt="A \in \schmC" title="A \in \schmC"></span>, define <span class="inline-formula"><img class="img-inline-formula img-formula" width="181" height="18" src="https://math.fontein.de/formulae/MjW8rUxKNSZB8J.yREADclOAssxM5vK3KeoVJA.svgz" alt="\eta_u(A) : h_X(A) \to H(A)" title="\eta_u(A) : h_X(A) \to H(A)"></span> by <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="18" src="https://math.fontein.de/formulae/7VWXfaNHcyUtSZkGdb.nMMpqwTKlbJeS_F_0oQ.svgz" alt="g \mapsto H(g)(u)" title="g \mapsto H(g)(u)"></span>. Now let <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="17" src="https://math.fontein.de/formulae/Y.AClLlF2mv0FNOKF1f6k7Thq3uaI93Q032lDQ.svgz" alt="\varphi : A \to A'" title="\varphi : A \to A'"></span> be a morphism; then we have the following diagram:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="540" height="303" src="https://math.fontein.de/formulae/SJ8EuLjBdE6uRDbXGvLYY.VppgAYSzQjwC0Snw.svgz" alt="\xymatrix{ g \ar@{|->}[rrr] \ar@{|->}[dddd] & & & g \circ \varphi \ar@{|->}[ddd] \\ & \Hom_\schmC(A, X) \ar@{=}[d] & \Hom_\schmC(A', X) \ar@{=}[d] & \\ & h_X(A) \ar[r]^{h_\varphi} \ar[d]_{\eta_u(A)} & h_X(A') \ar[d]^{\eta_u(A')} & \\ & H(A) \ar[r]_{H(\varphi)} & H(A') & H(g \circ \varphi)(u) \\ H(g)(u) \ar@{|->}[rr] & & H(\varphi)( H(g)(u) ) & }" title="\xymatrix{ g \ar@{|->}[rrr] \ar@{|->}[dddd] & & & g \circ \varphi \ar@{|->}[ddd] \\ & \Hom_\schmC(A, X) \ar@{=}[d] & \Hom_\schmC(A', X) \ar@{=}[d] & \\ & h_X(A) \ar[r]^{h_\varphi} \ar[d]_{\eta_u(A)} & h_X(A') \ar[d]^{\eta_u(A')} & \\ & H(A) \ar[r]_{H(\varphi)} & H(A') & H(g \circ \varphi)(u) \\ H(g)(u) \ar@{|->}[rr] & & H(\varphi)( H(g)(u) ) & }">
</div>
<p>
But since <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/592A9cpJqEapTTR0mQzYD2FBigGo_84WF1hl8w.svgz" alt="H" title="H"></span> is a contravariant functor, <span class="inline-formula"><img class="img-inline-formula img-formula" width="188" height="18" src="https://math.fontein.de/formulae/kXpN9Bm2Q.rVEyWzspwNI_kn7NfKTmDT4mQIYw.svgz" alt="H(g \circ \varphi) = H(\varphi) \circ H(g)" title="H(g \circ \varphi) = H(\varphi) \circ H(g)"></span>; therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="233" height="18" src="https://math.fontein.de/formulae/PkTGBYBjV7bC5lm5fNE_5sQlw4O2ObhZhidhMA.svgz" alt="H(\varphi)(H(g)(u)) = H(g \circ \varphi)(u)" title="H(\varphi)(H(g)(u)) = H(g \circ \varphi)(u)"></span>. This shows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="18" height="11" src="https://math.fontein.de/formulae/.QH_ZUJwqD06OUKylY5SdFUl1DxYiMtScGYpRg.svgz" alt="\eta_u" title="\eta_u"></span> is a natural transformation, i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="21" src="https://math.fontein.de/formulae/4brkD0skG.xh5dDnU40ErKbjF9KQ47EyrHOmow.svgz" alt="\eta_u \in \hat{\schmC}" title="\eta_u \in \hat{\schmC}"></span>.
</p>
<p>
Next, we have to show that
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="300" height="80" src="https://math.fontein.de/formulae/dE_HRhMtUnuwYTkE4c.f9TyfyhGf0SUBb8Yk5A.svgz" alt="\eta :{} & \bullet_2(\bullet_1) \to \Hom_{\hat{\schmC}}(h(\bullet_1), \bullet_2), \\
& (X, H) \mapsto \begin{cases} H(X) \to \Hom_{\hat{\schmC}}(h_X, H) \\ u \mapsto \eta_u \end{cases}" title="\eta :{} & \bullet_2(\bullet_1) \to \Hom_{\hat{\schmC}}(h(\bullet_1), \bullet_2), \\
& (X, H) \mapsto \begin{cases} H(X) \to \Hom_{\hat{\schmC}}(h_X, H) \\ u \mapsto \eta_u \end{cases}">
</div>
<p>
is a natural equivalence between functors
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="313" height="22" src="https://math.fontein.de/formulae/CbU0SvAn2mtKqKeGBD.qAsL_3moGMgenCiMCCQ.svgz" alt="\bullet_2(\bullet_1) : \schmC \times \hat{\schmC} \to \catSet, \quad (X, H) \mapsto H(X)" title="\bullet_2(\bullet_1) : \schmC \times \hat{\schmC} \to \catSet, \quad (X, H) \mapsto H(X)">
</div>
<p>
and
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="455" height="23" src="https://math.fontein.de/formulae/RwHhoAR9AcqyGoPHfXPP_V2Wu8ys1UFw7OhHWw.svgz" alt="\Hom_{\hat{\schmC}}(h(\bullet_1), \bullet_2) : \schmC \times \hat{\schmC} \to \catSet, \quad (X, H) \mapsto \Hom_{\hat{\schmC}}(h_X, H)." title="\Hom_{\hat{\schmC}}(h(\bullet_1), \bullet_2) : \schmC \times \hat{\schmC} \to \catSet, \quad (X, H) \mapsto \Hom_{\hat{\schmC}}(h_X, H).">
</div>
<p>
For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="17" src="https://math.fontein.de/formulae/4zTqqaRXVQiDUhsP4yhWpq1kK6q5hJaaAeTtDg.svgz" alt="X, X' \in \schmC" title="X, X' \in \schmC"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="21" src="https://math.fontein.de/formulae/3aLiuHzftdcHBH6SEi1.diFTLmYuJarS1N_.xA.svgz" alt="H, H' \in \hat{\schmC}" title="H, H' \in \hat{\schmC}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="17" src="https://math.fontein.de/formulae/cMSJpJBZ5j_mpOWNuAZ7CsgGppg8PM6j63_FGQ.svgz" alt="\varphi : X \to X'" title="\varphi : X \to X'"></span> a morphism and <span class="inline-formula"><img class="img-inline-formula img-formula" width="92" height="17" src="https://math.fontein.de/formulae/c_l6azpJjIhA.yxHyDivWsMiHWVTIFMxAZ_PCA.svgz" alt="\psi : H \to H'" title="\psi : H \to H'"></span> a natural transformation, and consider the following two diagrams:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="475" height="243" src="https://math.fontein.de/formulae/J8iRITrbLK4FFG0sXJzxkYnsUaaAEH3OqtDV1Q.svgz" alt="\xymatrix{ u \ar@{|-->}[ddd] \ar@{|-->}[rrr] & & & H(\varphi)(u) \ar@{|-->}[dd] \\ & H(X) \ar[r]^{H(\varphi)} \ar[d]_{\eta(X, H)} & H(X') \ar[d]^{\eta(X', H)} & \\ & \Hom_{\hat{\schmC}}(h_X, H) \ar[r]_{\Hom_{\hat{\schmC}}(h_\varphi, H)} & \Hom_{\hat{\schmC}}(h_{X'}, H) & \eta_{H(\varphi)(u)} \\ \eta_u \ar@{|-->}[rr] & & \eta_u \circ h_\varphi & }" title="\xymatrix{ u \ar@{|-->}[ddd] \ar@{|-->}[rrr] & & & H(\varphi)(u) \ar@{|-->}[dd] \\ & H(X) \ar[r]^{H(\varphi)} \ar[d]_{\eta(X, H)} & H(X') \ar[d]^{\eta(X', H)} & \\ & \Hom_{\hat{\schmC}}(h_X, H) \ar[r]_{\Hom_{\hat{\schmC}}(h_\varphi, H)} & \Hom_{\hat{\schmC}}(h_{X'}, H) & \eta_{H(\varphi)(u)} \\ \eta_u \ar@{|-->}[rr] & & \eta_u \circ h_\varphi & }">
</div>
<p>
Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="17" src="https://math.fontein.de/formulae/K8YPMqazUDQNBOhCS8n3iM8BFT5kF7WteJZjnQ.svgz" alt="\eta_u \circ h_\varphi" title="\eta_u \circ h_\varphi"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="14" src="https://math.fontein.de/formulae/TQnXqbbEwV.0Z9BvRHfCzCdQAqD1KGrjLkxXpw.svgz" alt="\eta_{H(\varphi)(u)}" title="\eta_{H(\varphi)(u)}"></span> are both natural transformations of functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="12" src="https://math.fontein.de/formulae/bt59FrZWLAmoBo23eX_WujC.WHTcaTd4Qpk3dg.svgz" alt="\schmC^{op} \to \catSet" title="\schmC^{op} \to \catSet"></span>. Hence, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/9tpiazplIU4lT9qQJZQvtmT.jkfAQYbAsv_qfQ.svgz" alt="A \in \schmC" title="A \in \schmC"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="203" height="18" src="https://math.fontein.de/formulae/3KdV0LSDkYUcg2ifX62l_kTnOGDY7hRCXW23Ew.svgz" alt="g \in h_X(A) = \Hom_\schmC(A, X)" title="g \in h_X(A) = \Hom_\schmC(A, X)"></span>. Then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="369" height="20" src="https://math.fontein.de/formulae/3TPxl9d4x14raPt0occRxK9k3GWcxVePBOxThw.svgz" alt="\eta_{H(\varphi)(u)}(A)(g) = H(g)( H(\varphi)(u) ) = H(\varphi \circ g)(u)" title="\eta_{H(\varphi)(u)}(A)(g) = H(g)( H(\varphi)(u) ) = H(\varphi \circ g)(u)">
</div>
<p>
as <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/592A9cpJqEapTTR0mQzYD2FBigGo_84WF1hl8w.svgz" alt="H" title="H"></span> is contravariant, and
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="517" height="18" src="https://math.fontein.de/formulae/jgr58pJaIbsPDNmRlPlAiv5_mYpQe6ldZFprdQ.svgz" alt="(\eta_u \circ h_\varphi)(A)(g) = \eta_u(A) \circ h_\varphi(A)(g) = \eta_u(A) \circ \varphi \circ g = H(\varphi \circ g)(u)." title="(\eta_u \circ h_\varphi)(A)(g) = \eta_u(A) \circ h_\varphi(A)(g) = \eta_u(A) \circ \varphi \circ g = H(\varphi \circ g)(u).">
</div>
<p>
Now consider the following diagram:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="476" height="241" src="https://math.fontein.de/formulae/0rxeiECujfHgRUP2KQtW2iUQtbhmJQ6kuf2XpA.svgz" alt="\xymatrix{ u \ar@{|-->}[ddd] \ar@{|-->}[rrr] & & & \psi(X)(u) \ar@{|-->}[dd] \\ & H(X) \ar[r]^{\psi(X)} \ar[d]_{\eta(X, H)} & H'(A) \ar[d]^{\eta(X, H')} & \\ & \Hom_{\hat{\schmC}}(h_X, H) \ar[r]_{\Hom_{\hat{\schmC}}(h_X, \psi)} & \Hom_{\hat{\schmC}}(h_X, H') & \eta_{\psi(X)(u)} \\ \eta_u \ar@{|-->}[rr] & & \psi \circ \eta_u & }" title="\xymatrix{ u \ar@{|-->}[ddd] \ar@{|-->}[rrr] & & & \psi(X)(u) \ar@{|-->}[dd] \\ & H(X) \ar[r]^{\psi(X)} \ar[d]_{\eta(X, H)} & H'(A) \ar[d]^{\eta(X, H')} & \\ & \Hom_{\hat{\schmC}}(h_X, H) \ar[r]_{\Hom_{\hat{\schmC}}(h_X, \psi)} & \Hom_{\hat{\schmC}}(h_X, H') & \eta_{\psi(X)(u)} \\ \eta_u \ar@{|-->}[rr] & & \psi \circ \eta_u & }">
</div>
<p>
Now <span class="inline-formula"><img class="img-inline-formula img-formula" width="47" height="16" src="https://math.fontein.de/formulae/Rw.zYs5_LRIshh8l_uGR7u5sEV_UrEcur0aRBQ.svgz" alt="\psi \circ \eta_u" title="\psi \circ \eta_u"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="14" src="https://math.fontein.de/formulae/1Vc3Jn1kzZbafDz18F5BfRNbzQK16AdDfGm7DQ.svgz" alt="\eta_{\psi(X)(u)}" title="\eta_{\psi(X)(u)}"></span> are both natural transformations of functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="12" src="https://math.fontein.de/formulae/bt59FrZWLAmoBo23eX_WujC.WHTcaTd4Qpk3dg.svgz" alt="\schmC^{op} \to \catSet" title="\schmC^{op} \to \catSet"></span>. Hence, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/9tpiazplIU4lT9qQJZQvtmT.jkfAQYbAsv_qfQ.svgz" alt="A \in \schmC" title="A \in \schmC"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="203" height="18" src="https://math.fontein.de/formulae/3KdV0LSDkYUcg2ifX62l_kTnOGDY7hRCXW23Ew.svgz" alt="g \in h_X(A) = \Hom_\schmC(A, X)" title="g \in h_X(A) = \Hom_\schmC(A, X)"></span>. Then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="383" height="20" src="https://math.fontein.de/formulae/EQHZrufPCEYajUU99pJCH2eki12pcs00hXpIEw.svgz" alt="\eta_{\psi(X)(u)}(g) = H'(g)( \psi(X)(u) ) = H'(g) \circ \psi(X)(u)" title="\eta_{\psi(X)(u)}(g) = H'(g)( \psi(X)(u) ) = H'(g) \circ \psi(X)(u)">
</div>
<p>
and
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="406" height="18" src="https://math.fontein.de/formulae/53FuHVQSujtRsipGM989tPJb1PJzO35RIsEIXQ.svgz" alt="(\psi \circ \eta_u)(A)(g) = \psi(A) \circ \eta_u(A)(g) = \psi(A) \circ H(g)(u)." title="(\psi \circ \eta_u)(A)(g) = \psi(A) \circ \eta_u(A)(g) = \psi(A) \circ H(g)(u).">
</div>
<p>
Hence, the condition that these two elements are the same is equivalent to the fact that the diagram
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="189" height="118" src="https://math.fontein.de/formulae/5uXBgWYg9xEf811K0OW8zG8fUgLNblB1ruI7qg.svgz" alt="\xymatrix{ H(X) \ar[d]_{\psi(X)} \ar[r]^{H(g)} & H(A) \ar[d]^{\psi(A)} \\ H'(X) \ar[r]_{H'(g)} & H'(A) }" title="\xymatrix{ H(X) \ar[d]_{\psi(X)} \ar[r]^{H(g)} & H(A) \ar[d]^{\psi(A)} \\ H'(X) \ar[r]_{H'(g)} & H'(A) }">
</div>
<p>
commutes; but that follows from the fact that <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="16" src="https://math.fontein.de/formulae/5YQhZpth_qEZfoZavf10MlpXOOEYGw_G8IF6hw.svgz" alt="\psi" title="\psi"></span> is a natural transformation <span class="inline-formula"><img class="img-inline-formula img-formula" width="65" height="13" src="https://math.fontein.de/formulae/SFqBTbFK7E45fYgjN38mjICtLtUDSvAHJOnyww.svgz" alt="H \to H'" title="H \to H'"></span>. Hence, we have shown that <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/LJX59haeTalq4bSnxWApTe0lMbaooEzjD2TZqQ.svgz" alt="\eta" title="\eta"></span> is a natural transformation.
</p>
<p>
It is left to show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/LJX59haeTalq4bSnxWApTe0lMbaooEzjD2TZqQ.svgz" alt="\eta" title="\eta"></span> is in fact an equivalence, i.e. that for <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="18" src="https://math.fontein.de/formulae/HY2.U1y7o.VaLzfw14uMyV1TzpRpbm55.fMqww.svgz" alt="H \in \hat{\schmC}" title="H \in \hat{\schmC}"></span>, the map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="340" height="19" src="https://math.fontein.de/formulae/CWKAYjEvoZNzk5WtoRrlyWJw_ZuSCXD2ExA4tQ.svgz" alt="\eta(X, H) : H(X) \to \Hom_{\hat{\schmC}}(h_X, H), \quad u \mapsto \eta_u" title="\eta(X, H) : H(X) \to \Hom_{\hat{\schmC}}(h_X, H), \quad u \mapsto \eta_u">
</div>
<p>
is bijective. First, we show that it is injective; for that, note that for <span class="inline-formula"><img class="img-inline-formula img-formula" width="78" height="18" src="https://math.fontein.de/formulae/s6FePyyeCb5DV497Xa7ypI6GaAAbgSDxQJRldw.svgz" alt="u \in H(X)" title="u \in H(X)"></span> we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="329" height="20" src="https://math.fontein.de/formulae/cN_D417dI9A9z7ZBwsFg9XxyJQcTer42RM940A.svgz" alt="\eta_u(X)(\id_X) = H(\id_X)(u) = \id_{H(X)}(u) = u" title="\eta_u(X)(\id_X) = H(\id_X)(u) = \id_{H(X)}(u) = u"></span>, whence <span class="inline-formula"><img class="img-inline-formula img-formula" width="18" height="11" src="https://math.fontein.de/formulae/.QH_ZUJwqD06OUKylY5SdFUl1DxYiMtScGYpRg.svgz" alt="\eta_u" title="\eta_u"></span> uniquely determines <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="8" src="https://math.fontein.de/formulae/Ei1EDDiFBqhkFoiwvRad2hcmf_KVutGWDniJpA.svgz" alt="u" title="u"></span>. To show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="18" src="https://math.fontein.de/formulae/N3FjzRsrjMWqAnr_uBni.j_Bo4HdGHQpTliEwg.svgz" alt="\eta(X, H)" title="\eta(X, H)"></span> is surjective, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="144" height="19" src="https://math.fontein.de/formulae/7s9WS5FWDK5ZqrSayEqUSD6epiPlRt_XI4htiA.svgz" alt="\eta' \in \Hom_{\hat{\schmC}}(h_X, H)" title="\eta' \in \Hom_{\hat{\schmC}}(h_X, H)"></span>. Set <span class="inline-formula"><img class="img-inline-formula img-formula" width="125" height="18" src="https://math.fontein.de/formulae/M1qDqf13vZYzx3OwUDNEwvQlPye0oBbkyki8tg.svgz" alt="u := \eta'(X)(\id_X)" title="u := \eta'(X)(\id_X)"></span>; if <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/9tpiazplIU4lT9qQJZQvtmT.jkfAQYbAsv_qfQ.svgz" alt="A \in \schmC" title="A \in \schmC"></span> is an object and <span class="inline-formula"><img class="img-inline-formula img-formula" width="203" height="18" src="https://math.fontein.de/formulae/3KdV0LSDkYUcg2ifX62l_kTnOGDY7hRCXW23Ew.svgz" alt="g \in h_X(A) = \Hom_\schmC(A, X)" title="g \in h_X(A) = \Hom_\schmC(A, X)"></span>, consider the following commutative diagram:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="460" height="239" src="https://math.fontein.de/formulae/H0P__iVxOW5fHunMUH4AAuVqUIqfZygesZfLFQ.svgz" alt="\xymatrix{ \id_X \ar@{|->}[rrr] \ar@{|->}[ddd] & & & g \ar@{|->}[dd] \\ & \Hom_\schmC(X, X) \ar[r]^{\Hom_\schmC(g, X)} \ar[d]_{\eta'(X)} & \Hom_\schmC(A, X) \ar[d]^{\eta'(A)} & \\ & H(X) \ar[r]_{H(g)} & H(A) & \eta'(A)(g) \\ u \ar@{|->}[rr] & & H(g)(u) & }" title="\xymatrix{ \id_X \ar@{|->}[rrr] \ar@{|->}[ddd] & & & g \ar@{|->}[dd] \\ & \Hom_\schmC(X, X) \ar[r]^{\Hom_\schmC(g, X)} \ar[d]_{\eta'(X)} & \Hom_\schmC(A, X) \ar[d]^{\eta'(A)} & \\ & H(X) \ar[r]_{H(g)} & H(A) & \eta'(A)(g) \\ u \ar@{|->}[rr] & & H(g)(u) & }">
</div>
<p>
Hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="242" height="18" src="https://math.fontein.de/formulae/jX0iYMACfkvdMyIERCllWKbAebYLbG4Kt5Nyaw.svgz" alt="\eta'(A)(g) = H(g)(u) = \eta_u(A)(g)" title="\eta'(A)(g) = H(g)(u) = \eta_u(A)(g)"></span>. Since <span class="inline-formula"><img class="img-inline-formula img-formula" width="13" height="12" src="https://math.fontein.de/formulae/AdTxuIawp9Z_8j4FFnLDOtejF7gGMlzudPQwOA.svgz" alt="A" title="A"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/UxCAbnAvo.LhIVRUfLjfwIlPAfhwrVfYLg_Adw.svgz" alt="g" title="g"></span> were arbitrary, we get <span class="inline-formula"><img class="img-inline-formula img-formula" width="56" height="17" src="https://math.fontein.de/formulae/mCs8D2BOYFAwAWba_JD5oWXZY_1m4cQgZk3QNw.svgz" alt="\eta' = \eta_u" title="\eta' = \eta_u"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Wow, looks like a huge collection of <a href="https://en.wikipedia.org/wiki/Abstract_nonsense">abstract nonsense</a>, eh? Well, the good news is that the worst part is done. Now, let us prove the corollary.
</p>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof (of the <a href="https://math.fontein.de/2009/08/16/fun-with-representable-functors-or-why-i-like-yondeas-lemma/#yoneda-corollary">Corollary</a>).
</div>
<div class="theorem-content theorem-proof-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="16" src="https://math.fontein.de/formulae/nn_Qx0YGNdNQ.McKxQQe0g.e4iyvVCkR.LS4Ng.svgz" alt="X, Y \in \schmC" title="X, Y \in \schmC"></span>, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="20" src="https://math.fontein.de/formulae/BfyFTMJt6qtbkoQDRcO2dZ5Pgl5yJGXsjkHeWg.svgz" alt="H := h_Y \in \hat{\schmC}" title="H := h_Y \in \hat{\schmC}"></span>; then <span class="inline-formula"><img class="img-inline-formula img-formula" width="168" height="18" src="https://math.fontein.de/formulae/Qcsv5gklzjnE_gdOPGM5nnoTH.NPw40oF7ncng.svgz" alt="H(X) = \Hom_\schmC(X, Y)" title="H(X) = \Hom_\schmC(X, Y)"></span>, and by <a href="https://math.fontein.de/2009/08/16/fun-with-representable-functors-or-why-i-like-yondeas-lemma/#yoneda-lemma">Yoneda's Lemma</a>, the map
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="519" height="19" src="https://math.fontein.de/formulae/tS6TwNlqIovVEWwzGk725BEpzIm_kSi_e9rkBg.svgz" alt="\eta(X, H) : \Hom_\schmC(X, Y) = H(X) \to \Hom_{\hat{\schmC}}(h_X, H) = \Hom_{\hat{\schmC}}(h_X, h_Y)" title="\eta(X, H) : \Hom_\schmC(X, Y) = H(X) \to \Hom_{\hat{\schmC}}(h_X, H) = \Hom_{\hat{\schmC}}(h_X, h_Y)">
</div>
<p>
is bijective. We have to check that <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="18" src="https://math.fontein.de/formulae/N3FjzRsrjMWqAnr_uBni.j_Bo4HdGHQpTliEwg.svgz" alt="\eta(X, H)" title="\eta(X, H)"></span> equals the map <span class="inline-formula"><img class="img-inline-formula img-formula" width="265" height="19" src="https://math.fontein.de/formulae/KKoumkTafaIlgyvlwsi24opj2yXIzPMvTHHQhQ.svgz" alt="h : \Hom_\schmC(X, Y) \to \Hom_{\hat{\schmC}}(h_X, h_Y)" title="h : \Hom_\schmC(X, Y) \to \Hom_{\hat{\schmC}}(h_X, h_Y)"></span>; for that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="83" height="12" src="https://math.fontein.de/formulae/D6XDU.KQedkHLZio9rwedUSnCVPShYHdNJoYRg.svgz" alt="u : X \to Y" title="u : X \to Y"></span> be a morphism, <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/9tpiazplIU4lT9qQJZQvtmT.jkfAQYbAsv_qfQ.svgz" alt="A \in \schmC" title="A \in \schmC"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/mEXTAecG9bdu5.f72xqv9l5iXmDrva2j0orOBQ.svgz" alt="g \in h_X(A)" title="g \in h_X(A)"></span>. Then
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="483" height="44" src="https://math.fontein.de/formulae/ewYXzUS06iJO0UXaPQRbotDACLM7tiyR.gFe6Q.svgz" alt="h(u)(A)(g) ={} & h_u(A)(g) = \Hom_\schmC(A, u)(g) = u \circ g \\
{}={} & \Hom_\schmC(g, Y)(u) = h_Y(g)(u) = \eta(X, H)(u)(A)(g)." title="h(u)(A)(g) ={} & h_u(A)(g) = \Hom_\schmC(A, u)(g) = u \circ g \\
{}={} & \Hom_\schmC(g, Y)(u) = h_Y(g)(u) = \eta(X, H)(u)(A)(g).">
</div>
<p>
Finally, we have to show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span> is injective on objects. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="177" height="18" src="https://math.fontein.de/formulae/6iUDXKnxub2mJ2dbiDp5cnK8oUaTCn2UpnBNtQ.svgz" alt="h_X(X) = \Hom_\schmC(X, X)" title="h_X(X) = \Hom_\schmC(X, X)"></span>. Since morphism sets for distinct objects are assumed to be disjunct, it follows that we can get <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/El54jcSPwk.2ASHD5GrpJ57pTPCUt4gRfS4OSg.svgz" alt="X" title="X"></span> back from <span class="inline-formula"><img class="img-inline-formula img-formula" width="54" height="18" src="https://math.fontein.de/formulae/pCzr.L4B4wC1CPzHWPPGJPTKqKZ7T0.c8bxDMw.svgz" alt="h_X(X)" title="h_X(X)"></span>.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
Well. After all this abstract nonsense, let us do some more concrete abstract nonsense. First, a lengthy definition which will turn out to be quite cool.
</p>
<div class="theorem-environment theorem-definition-environment">
<div class="theorem-header theorem-definition-header">
Definition.
</div>
<div class="theorem-content theorem-definition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> be a category with a <a href="https://en.wikipedia.org/wiki/Final_object">final object</a> <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> in which <a href="https://en.wikipedia.org/wiki/Category_product">finite products</a> <span class="inline-formula"><img class="img-inline-formula img-formula" width="49" height="14" src="https://math.fontein.de/formulae/vjLFba8tUZAiezgF0vGV0K.Qqk3FB2nvORaQyg.svgz" alt="G \times G" title="G \times G"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="14" src="https://math.fontein.de/formulae/HTWoDxIQhzVrOBJbQRHgSmLscQT3R7iiVy4olw.svgz" alt="G \times G \times G" title="G \times G \times G"></span> exist for all objects <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/Vw3zO2QD64b3PmwrbOuK1vgho1W_7TCKegZoig.svgz" alt="G \in \schmC" title="G \in \schmC"></span>.
</p>
<ol class="enum-level-1">
<li>
<p>
An object <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/Vw3zO2QD64b3PmwrbOuK1vgho1W_7TCKegZoig.svgz" alt="G \in \schmC" title="G \in \schmC"></span> together with morphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="14" src="https://math.fontein.de/formulae/NMqsbAaH46PqOWuFBUwRZmOEPxV441fKQyRRww.svgz" alt="m : G \times G \to G" title="m : G \times G \to G"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="12" src="https://math.fontein.de/formulae/sOaXQ02qOl_MkP6vwTXZqaJXjDYqRF5qOWBnwg.svgz" alt="i : G \to G" title="i : G \to G"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="12" src="https://math.fontein.de/formulae/eQjeUCX4ZxzafGojzGu5E4_MePxVGfUVb14jRw.svgz" alt="e : S \to G" title="e : S \to G"></span> is called a <a href="https://en.wikipedia.org/wiki/Group_object">group object</a> if the following diagrams commute:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="295" height="104" src="https://math.fontein.de/formulae/.ztIPSfrfmkZzWQkhXA19DE3vyaO0l88P_T_cQ.svgz" alt="\xymatrix@C+1cm{ G \times G \times G \ar[r]^{m \times \id_G} \ar[d]_{\qquad \id_G \times m} & G \times G \ar[d]^m \\ G \times G \ar[r]_m & G }" title="\xymatrix@C+1cm{ G \times G \times G \ar[r]^{m \times \id_G} \ar[d]_{\qquad \id_G \times m} & G \times G \ar[d]^m \\ G \times G \ar[r]_m & G }">
</div>
<div class="display-formula">
<img class="img-display-formula img-formula" width="340" height="105" src="https://math.fontein.de/formulae/BDa9Wfeo.C5AImSmtfxyHaK4lihxlu._owokbw.svgz" alt="\xymatrix@C+0.5cm{ G \ar[r]^{(\id_G, i) \qquad} \ar[d] & G \times G \ar[d]^m \\ S \ar[r]_e & G } \qquad \xymatrix@C+0.5cm{ G \ar[r]^{(i, \id_G) \qquad} \ar[d] & G \times G \ar[d]^m \\ S \ar[r]_e & G }" title="\xymatrix@C+0.5cm{ G \ar[r]^{(\id_G, i) \qquad} \ar[d] & G \times G \ar[d]^m \\ S \ar[r]_e & G } \qquad \xymatrix@C+0.5cm{ G \ar[r]^{(i, \id_G) \qquad} \ar[d] & G \times G \ar[d]^m \\ S \ar[r]_e & G }">
</div>
<div class="display-formula">
<img class="img-display-formula img-formula" width="390" height="97" src="https://math.fontein.de/formulae/Qq48umNQgUggr97GHeRAnFvu3glMyULyMWBFSA.svgz" alt="\xymatrix@C-0.5cm{ G \times S \ar[rr]^{\id_G \times e} \ar[dr]_\cong & & G \times G \ar[dl]^m \\ & G & } \qquad \xymatrix@C-0.5cm{ S \times G \ar[rr]^{e \times \id_G} \ar[dr]_\cong & & G \times G \ar[dl]^m \\ & G & }" title="\xymatrix@C-0.5cm{ G \times S \ar[rr]^{\id_G \times e} \ar[dr]_\cong & & G \times G \ar[dl]^m \\ & G & } \qquad \xymatrix@C-0.5cm{ S \times G \ar[rr]^{e \times \id_G} \ar[dr]_\cong & & G \times G \ar[dl]^m \\ & G & }">
</div>
</li>
<li>We say that a group object <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/xVBTg1pR3BXujqwBzTaHu8l8jUmSZxFPDwxJLQ.svgz" alt="(G, m, i, e)" title="(G, m, i, e)"></span> is <em>commutative</em> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="8" src="https://math.fontein.de/formulae/WsKWMUu28.hceFh9sRb9QrVznyageCBrz8LNMQ.svgz" alt="m \circ w = m" title="m \circ w = m"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="154" height="14" src="https://math.fontein.de/formulae/WPuwqebtftlO2luhnEcJyG3d7jZ3CfAazdI6Zw.svgz" alt="w : G \times G \to G \times G" title="w : G \times G \to G \times G"></span> switches its operands.</li>
<li>Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="117" height="18" src="https://math.fontein.de/formulae/3ino0gNUAwD2qcS12WSqa26gc8ElK_A_pxmhYw.svgz" alt="(G, m_G, i_G, e_G)" title="(G, m_G, i_G, e_G)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="123" height="18" src="https://math.fontein.de/formulae/JAQ3tAv1FtLS_tOT.9IWrnjB6HuH7ORfDAtY9A.svgz" alt="(H, m_H, i_H, e_H)" title="(H, m_H, i_H, e_H)"></span> be group objects. A morphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="16" src="https://math.fontein.de/formulae/0ZcSf_C.xN1ExNRQp1v2u3USVhsII3WMlgLYRA.svgz" alt="\varphi : G \to H" title="\varphi : G \to H"></span> is called a <em>homomorphism of group objects</em> <span class="inline-formula"><img class="img-inline-formula img-formula" width="294" height="18" src="https://math.fontein.de/formulae/BRcIXhiGIFZVtK7plEdsIC2BL8ogC6M09LgPQw.svgz" alt="\varphi : (G, m_G, i_G, e_G) \to (H, m_H, i_H, e_H)" title="\varphi : (G, m_G, i_G, e_G) \to (H, m_H, i_H, e_H)"></span> if <span class="inline-formula"><img class="img-inline-formula img-formula" width="184" height="18" src="https://math.fontein.de/formulae/JuvJdN8caCWK4BM1ee22elDY48_JVIgwdAs.jQ.svgz" alt="m_H \circ (\varphi \times \varphi) = \varphi \circ m_G" title="m_H \circ (\varphi \times \varphi) = \varphi \circ m_G"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="118" height="15" src="https://math.fontein.de/formulae/41YOqExJ72Dgx4HI7TR1yC7IE7G1sinPwvUnAw.svgz" alt="i_H \circ \varphi = \varphi \circ i_G" title="i_H \circ \varphi = \varphi \circ i_G"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="11" src="https://math.fontein.de/formulae/Zwb1hF7EEqZ4sZOt0EUS5DzDcXXSzCSfxQ8Kfg.svgz" alt="e_H = \varphi \circ e_G" title="e_H = \varphi \circ e_G"></span>.</li>
</ol>
</div>
</div>
<p>
Let us first give some examples:
</p>
<div class="theorem-environment theorem-example-environment">
<div class="theorem-header theorem-example-header">
Example.
</div>
<div class="theorem-content theorem-example-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="12" src="https://math.fontein.de/formulae/TfkE2mxa_cnslvdn73NVO7sV3O3PG.QGFfl1tA.svgz" alt="\schmC = \catSet" title="\schmC = \catSet"></span>; then <span class="inline-formula"><img class="img-inline-formula img-formula" width="27" height="18" src="https://math.fontein.de/formulae/DhzheRFBT08rqjhqIUTW8woIvkklU1DCZ1q.HQ.svgz" alt="\{ 1 \}" title="\{ 1 \}"></span> is a final object in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span>. The group objects in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> are exactly the groups. A group object is commutative if the corresponding group is commutative, and homomorphisms of group objects are nothing else than group homomorphisms.
</p>
</div>
</div>
<div class="theorem-environment theorem-example-environment">
<div class="theorem-header theorem-example-header">
Example.
</div>
<div class="theorem-content theorem-example-content">
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> is the category of topological spaces, with continuous maps as morphisms, then the group objects in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> are exactly the <a href="https://en.wikipedia.org/wiki/Topological_group">topological groups</a>. Homomorphisms of group objects are again nothing else than continuous group homomorphisms.
</p>
</div>
</div>
<div class="theorem-environment theorem-example-environment">
<div class="theorem-header theorem-example-header">
Example.
</div>
<div class="theorem-content theorem-example-content">
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="59" height="16" src="https://math.fontein.de/formulae/bAH4iPq.Rv9s8yV0crKIw6WXaJkxc4SOAwN0FQ.svgz" alt="\schmC = \catGrp" title="\schmC = \catGrp"></span> is the category of all groups, the group objects in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> are a bit more interesting. A final object <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> is given by <span class="inline-formula"><img class="img-inline-formula img-formula" width="89" height="18" src="https://math.fontein.de/formulae/G8AQTw_QWkHVDoAl_DweXIYM1_56JEQ_XrV8dA.svgz" alt="S = (\{ 1 \}, \cdot)" title="S = (\{ 1 \}, \cdot)"></span>, a group of one element (in fact, <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span> is an initial object as well).
</p>
<p>
First, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="18" src="https://math.fontein.de/formulae/N.g_ZDRV64ClKP6cGg.VBCAXUSsDSIzyv.qsMQ.svgz" alt="(A, \cdot)" title="(A, \cdot)"></span> be an abelian group. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="119" height="14" src="https://math.fontein.de/formulae/PNbVuHZmULJwT1fRKf0vsO7goa607NQ9SYplCA.svgz" alt="m : A \times A \to A" title="m : A \times A \to A"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="18" src="https://math.fontein.de/formulae/U9tpnyxUHMtz1VKECYBVNDiFBuFeSNeDKkOCtw.svgz" alt="(x, x') \mapsto x x'" title="(x, x') \mapsto x x'"></span> is a group homomorphism. So is <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="12" src="https://math.fontein.de/formulae/USF8cItYTsoLsenYsg51a8nDYi5jbX96SnmYlw.svgz" alt="i : A \to A" title="i : A \to A"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="67" height="14" src="https://math.fontein.de/formulae/.AD02LsYgHXW2JIm8b0reAIZgDxVJ4LSsP9YGA.svgz" alt="x \mapsto x^{-1}" title="x \mapsto x^{-1}"></span>, and the map <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="12" src="https://math.fontein.de/formulae/BUWUrksfcgLkQJC5GljiwsmeC.0ObBQ25nISdw.svgz" alt="e : S \to A" title="e : S \to A"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="14" src="https://math.fontein.de/formulae/Jw3ACsDNOPArxIkWWQlq7NU9JcLPg7PVHFJ9Cg.svgz" alt="1 \mapsto 1_A" title="1 \mapsto 1_A"></span> is a group homomorphism as well. Hence, every abelian group <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="18" src="https://math.fontein.de/formulae/N.g_ZDRV64ClKP6cGg.VBCAXUSsDSIzyv.qsMQ.svgz" alt="(A, \cdot)" title="(A, \cdot)"></span> gives rise to a commutative group object <span class="inline-formula"><img class="img-inline-formula img-formula" width="107" height="18" src="https://math.fontein.de/formulae/jfvGr4ghrc7Lr88inXGPFQVc5jWqBg2MP3tgow.svgz" alt="((A, \cdot), m, i, e)" title="((A, \cdot), m, i, e)"></span>.
</p>
<p>
Now, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="108" height="18" src="https://math.fontein.de/formulae/L_zqXJu409V.djTB9vyqgPD46cPjtLnygRIcSQ.svgz" alt="((G, \cdot), m, i, e)" title="((G, \cdot), m, i, e)"></span> be a group object. As <span class="inline-formula"><img class="img-inline-formula img-formula" width="8" height="8" src="https://math.fontein.de/formulae/5lacGHrKTIknHRnPj9mi0l0XngQ0jMuai6e_.Q.svgz" alt="e" title="e"></span> is a group homomorphism, we have <span class="inline-formula"><img class="img-inline-formula img-formula" width="75" height="18" src="https://math.fontein.de/formulae/J6NT5o_F0b6se6AqVwv1T3oaNIJ1aWnCanTV3A.svgz" alt="e(1) = 1_G" title="e(1) = 1_G"></span>. Next, since <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="14" src="https://math.fontein.de/formulae/NMqsbAaH46PqOWuFBUwRZmOEPxV441fKQyRRww.svgz" alt="m : G \times G \to G" title="m : G \times G \to G"></span> is a group homomorphism, we get
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="239" height="18" src="https://math.fontein.de/formulae/E_RxBIigBdOKoq_75Lmk85jB7wzF1Zay1xQbiw.svgz" alt="m(g, h) = m(g, 1) m(1, h) = g h," title="m(g, h) = m(g, 1) m(1, h) = g h,">
</div>
<p>
i.e. <span class="inline-formula"><img class="img-inline-formula img-formula" width="44" height="8" src="https://math.fontein.de/formulae/Q.PkZnGSbuRbXWD3Q1GPXBO9kD6QPWnaI8RQ1g.svgz" alt="m = \cdot" title="m = \cdot"></span>. Now, <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="12" src="https://math.fontein.de/formulae/sOaXQ02qOl_MkP6vwTXZqaJXjDYqRF5qOWBnwg.svgz" alt="i : G \to G" title="i : G \to G"></span> must be a group homomorphism as well, and the group object axioms force <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="19" src="https://math.fontein.de/formulae/kO1KQcvq8X.ClmnYRxTuCyTpy_NH1ebWirHbYw.svgz" alt="i(g) = g^{-1}" title="i(g) = g^{-1}"></span> for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="45" height="16" src="https://math.fontein.de/formulae/gJ.o_bci_fOa2j72N9wbYH666wJAQXgkwK7CBg.svgz" alt="g \in G" title="g \in G"></span>; hence, for all <span class="inline-formula"><img class="img-inline-formula img-formula" width="63" height="16" src="https://math.fontein.de/formulae/pgQNtO5E6gJdFMDFE7A8n3ASmBLfVUqUqXYueg.svgz" alt="g, h \in G" title="g, h \in G"></span>,
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="402" height="20" src="https://math.fontein.de/formulae/0l9k2ILr13uerS_oiWVbe0nHJVdVoBsYaaVRoQ.svgz" alt="g h = i(g^{-1}) i(h^{-1}) = i(g^{-1} h^{-1}) = (g^{-1} h^{-1})^{-1} = h g." title="g h = i(g^{-1}) i(h^{-1}) = i(g^{-1} h^{-1}) = (g^{-1} h^{-1})^{-1} = h g.">
</div>
<p>
Therefore, <span class="inline-formula"><img class="img-inline-formula img-formula" width="108" height="18" src="https://math.fontein.de/formulae/L_zqXJu409V.djTB9vyqgPD46cPjtLnygRIcSQ.svgz" alt="((G, \cdot), m, i, e)" title="((G, \cdot), m, i, e)"></span> forces <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="18" src="https://math.fontein.de/formulae/K2sYP9rMj7El42dxUZoIWXghBbrYW6Xl3TQq9Q.svgz" alt="(G, \cdot)" title="(G, \cdot)"></span> to be abelian, and the argument shows that <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="8" src="https://math.fontein.de/formulae/Ln3FmQu8Rxbv_tjKkeMJZrLhqAFAWD18KRPi9w.svgz" alt="m" title="m"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="6" height="12" src="https://math.fontein.de/formulae/S43oPTrFqmoVC.yqOcgzvrroaMU3pS7pa40ROQ.svgz" alt="i" title="i"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="8" height="8" src="https://math.fontein.de/formulae/5lacGHrKTIknHRnPj9mi0l0XngQ0jMuai6e_.Q.svgz" alt="e" title="e"></span> are uniquely determined by <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="18" src="https://math.fontein.de/formulae/K2sYP9rMj7El42dxUZoIWXghBbrYW6Xl3TQq9Q.svgz" alt="(G, \cdot)" title="(G, \cdot)"></span>.
</p>
<p>
Therefore, the group objects in <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="16" src="https://math.fontein.de/formulae/fuj0xeAJAbJYHrp_4.B.un8wkA7ZYPBfJvX47w.svgz" alt="\catGrp" title="\catGrp"></span> are exactly the abelian groups, and all of them are commutative. The group object homomorphisms between two abelian groups (with their only possible group object structure) are exactly the morphisms in <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="16" src="https://math.fontein.de/formulae/fuj0xeAJAbJYHrp_4.B.un8wkA7ZYPBfJvX47w.svgz" alt="\catGrp" title="\catGrp"></span> between the two objects.
</p>
</div>
</div>
<p>
Now, one can do other stuff with group objects <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/wolCU._XCwGlaxFD4lZCcDS1ODaeXVFpLnGvcg.svgz" alt="(G, m, e, i)" title="(G, m, e, i)"></span>. Namely, given any other object <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, we can turn <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="18" src="https://math.fontein.de/formulae/wO0ye2YjUE3eY2wtZMd4rXq5oWZvzz8_FmJj8Q.svgz" alt="\Hom_\schmC(X, G)" title="\Hom_\schmC(X, G)"></span> into a group:
</p>
<div class="theorem-environment theorem-proposition-environment">
<div class="theorem-header theorem-proposition-header">
Proposition.
</div>
<div class="theorem-content theorem-proposition-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/wolCU._XCwGlaxFD4lZCcDS1ODaeXVFpLnGvcg.svgz" alt="(G, m, e, i)" title="(G, m, e, i)"></span> be a group object in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span>. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, set <span class="inline-formula"><img class="img-inline-formula img-formula" width="142" height="18" src="https://math.fontein.de/formulae/Deb9XZTBSn03.K7ZPoJ8GaF_Tasi7CSg7Jciug.svgz" alt="H := \Hom_\schmC(X, G)" title="H := \Hom_\schmC(X, G)"></span> and define
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="325" height="43" src="https://math.fontein.de/formulae/I6jF8SFxrAT0W0Jv64isoVOTSBAIK4yMwnCoCg.svgz" alt="m_H :{} & H \times H \to H, \quad (f, g) \mapsto m \circ (f \times g), \\
i_H :{} & H \to H, \quad f \mapsto i \circ f" title="m_H :{} & H \times H \to H, \quad (f, g) \mapsto m \circ (f \times g), \\
i_H :{} & H \to H, \quad f \mapsto i \circ f">
</div>
<p>
and set <span class="inline-formula"><img class="img-inline-formula img-formula" width="99" height="11" src="https://math.fontein.de/formulae/jNRxpV.vrqZAWPlfCdM_s6n.iLIB_2MloN_wdQ.svgz" alt="e_H := e \circ \pi_X" title="e_H := e \circ \pi_X"></span>, where <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="15" src="https://math.fontein.de/formulae/wrxmunnGaRxO0GjUY0g.aNdiIlM4Q_gMdsxk9w.svgz" alt="\pi_X : X \to S" title="\pi_X : X \to S"></span> is the unique morphism to the final object <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span>. Then <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="18" src="https://math.fontein.de/formulae/fz.zKPyJsQRuegPik7o4JKGJGwkX9eIbtd2wHA.svgz" alt="(H, m_H)" title="(H, m_H)"></span> is a group whose inverses are given by <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="14" src="https://math.fontein.de/formulae/PO3B9J56nHGBJWi0AqT.AhBVNTTEKuuqKDUVyg.svgz" alt="i_H" title="i_H"></span> and whose neutral element is <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="10" src="https://math.fontein.de/formulae/M7rn22Iye5Ndpuu0F.VSttqwRHgZnD8XOP6LJQ.svgz" alt="e_H" title="e_H"></span>. If <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/wolCU._XCwGlaxFD4lZCcDS1ODaeXVFpLnGvcg.svgz" alt="(G, m, e, i)" title="(G, m, e, i)"></span> is commutative, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="18" src="https://math.fontein.de/formulae/fz.zKPyJsQRuegPik7o4JKGJGwkX9eIbtd2wHA.svgz" alt="(H, m_H)" title="(H, m_H)"></span> is abelian.
</p>
</div>
</div>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<p>
First, we show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="29" height="10" src="https://math.fontein.de/formulae/onreAD4If_aM_PF44Ny04Y0joMDEfR3JuN0YtA.svgz" alt="m_H" title="m_H"></span> is associative. For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="164" height="18" src="https://math.fontein.de/formulae/tZW98qzqJqDtoLW.d6JKuLi3Y_pCM1bYGZoiHg.svgz" alt="f, g, h \in \Hom_\schmC(X, G)" title="f, g, h \in \Hom_\schmC(X, G)"></span>. Then
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="378" height="97" src="https://math.fontein.de/formulae/pf1RBBebrT.nnntI4zZwnxjkAJf5JuTFexiIMA.svgz" alt="m_H(m_H(f, g), h) ={} & m_H(m \circ (f \times g), h) \\
{}={} & m \circ ((m \circ (f \times g)) \times h) \\
\text{and} \quad m_H(f, m_H(g, h)) ={} & m_H(f, m \circ (g \times h)) \\
{}={} & m \circ (f \times (m \circ (g \times h))." title="m_H(m_H(f, g), h) ={} & m_H(m \circ (f \times g), h) \\
{}={} & m \circ ((m \circ (f \times g)) \times h) \\
\text{and} \quad m_H(f, m_H(g, h)) ={} & m_H(f, m \circ (g \times h)) \\
{}={} & m \circ (f \times (m \circ (g \times h)).">
</div>
<p>
We can rewrite this to
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="413" height="44" src="https://math.fontein.de/formulae/KCF1GefRUaIfEWD9xBafpmq4K5VOYNQDyDeYnA.svgz" alt="m_H(m_H(f, g), h) ={} & m \circ (m \times \id_G) \circ (f \times g \times h) \\
\text{and} \quad m_H(f, m_H(g, h)) ={} & m \circ (\id_G \times m) \circ (f \times g \times h)," title="m_H(m_H(f, g), h) ={} & m \circ (m \times \id_G) \circ (f \times g \times h) \\
\text{and} \quad m_H(f, m_H(g, h)) ={} & m \circ (\id_G \times m) \circ (f \times g \times h),">
</div>
<p>
but from the definition of a group object, we know <span class="inline-formula"><img class="img-inline-formula img-formula" width="239" height="18" src="https://math.fontein.de/formulae/GsTGS5a54CQci70chby7heRuhLBw8sFw5y9pMw.svgz" alt="m \circ (m \times \id_G) = m \circ (\id_G \times m)" title="m \circ (m \times \id_G) = m \circ (\id_G \times m)"></span>. Therefore, both expressions are the same.
</p>
<p>
Next, let us show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="10" src="https://math.fontein.de/formulae/M7rn22Iye5Ndpuu0F.VSttqwRHgZnD8XOP6LJQ.svgz" alt="e_H" title="e_H"></span> is a neutral element. For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="130" height="18" src="https://math.fontein.de/formulae/iDxxWC4GDVg5oGNGcVPkZLRJ76iMD4VkWVYhPg.svgz" alt="f \in \Hom_\schmC(X, G)" title="f \in \Hom_\schmC(X, G)"></span>. Then
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="421" height="44" src="https://math.fontein.de/formulae/9NgCkgpiXWitAN49KlpR0huTC0_N5.pNcQ6tww.svgz" alt="m_H(f, e_H) ={} & m \circ (f \times e_H) = m \circ ((\id_G \circ f) \times (e \circ \pi_X)) \\
{}={} & m \circ (\id_G \times e) \circ (f \times \pi_X)" title="m_H(f, e_H) ={} & m \circ (f \times e_H) = m \circ ((\id_G \circ f) \times (e \circ \pi_X)) \\
{}={} & m \circ (\id_G \times e) \circ (f \times \pi_X)">
</div>
<p>
and, analogous, <span class="inline-formula"><img class="img-inline-formula img-formula" width="295" height="18" src="https://math.fontein.de/formulae/kONADPuSiIYOccNJQmMMeECnclwq1GaaaJvXYQ.svgz" alt="m_H(e_H, f) = m \circ (e \times \id_G) \circ (\pi_X \times f)" title="m_H(e_H, f) = m \circ (e \times \id_G) \circ (\pi_X \times f)"></span>. But from the commutative diagrams of the definition of a group object, it turns out that both are the same <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="16" src="https://math.fontein.de/formulae/M.ji_f0zsVnTLyaKegBkRJLOeqrrt6brNnapOQ.svgz" alt="f" title="f"></span>.
</p>
<p>
Now let us show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="14" src="https://math.fontein.de/formulae/PO3B9J56nHGBJWi0AqT.AhBVNTTEKuuqKDUVyg.svgz" alt="i_H" title="i_H"></span> is the inverse map. For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="130" height="18" src="https://math.fontein.de/formulae/iDxxWC4GDVg5oGNGcVPkZLRJ76iMD4VkWVYhPg.svgz" alt="f \in \Hom_\schmC(X, G)" title="f \in \Hom_\schmC(X, G)"></span>. Then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="400" height="18" src="https://math.fontein.de/formulae/Dldlav9bEO_efFOeViCrM8SSjYvwECUTdryQCg.svgz" alt="m_H(f, i_H(f)) = m \circ (f \times (i \circ f)) = m \circ (\id_G \times i) \circ f" title="m_H(f, i_H(f)) = m \circ (f \times (i \circ f)) = m \circ (\id_G \times i) \circ f">
</div>
<p>
and <span class="inline-formula"><img class="img-inline-formula img-formula" width="256" height="18" src="https://math.fontein.de/formulae/R1H0hq3tiXvcRuF7KVCFESlWuQgxB7VSpsaWyQ.svgz" alt="m_H(i_H(f), f) = m \circ (i \times \id_G) \circ f" title="m_H(i_H(f), f) = m \circ (i \times \id_G) \circ f"></span>. Now the definition of a group object gives <span class="inline-formula"><img class="img-inline-formula img-formula" width="290" height="18" src="https://math.fontein.de/formulae/VBSEHAYGP_jzm1s9hOBW4Gw8y7iyAzJyRjPeQA.svgz" alt="m \circ (\id_G \times i) = m \circ (i \times \id_G) = e \circ \pi_G" title="m \circ (\id_G \times i) = m \circ (i \times \id_G) = e \circ \pi_G"></span>, whence
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="453" height="18" src="https://math.fontein.de/formulae/cKFdnAOJD.IhjSw7kTpbkUKpgiRmJb.B__y1xQ.svgz" alt="m_H(f, i_H(f)) = m_H(i_H(f), f) = e \circ \pi_G \circ f = e \circ \pi_X = e_H." title="m_H(f, i_H(f)) = m_H(i_H(f), f) = e \circ \pi_G \circ f = e \circ \pi_X = e_H.">
</div>
<p>
Finally, assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/xVBTg1pR3BXujqwBzTaHu8l8jUmSZxFPDwxJLQ.svgz" alt="(G, m, i, e)" title="(G, m, i, e)"></span> is commutative. Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="18" src="https://math.fontein.de/formulae/Yq4N6F9zKYo3ZPr01efnHtQ8vuozgTUQpQkaBQ.svgz" alt="f, g \in \Hom_\schmC(X, G)" title="f, g \in \Hom_\schmC(X, G)"></span>; then
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="306" height="44" src="https://math.fontein.de/formulae/xy5bVKQ2uwaJi5r..Fe7lrAMGJrpe1Hf7ryBtg.svgz" alt="m_H(f, g) ={} & m \circ (f \times g) = m \circ w \circ (f, g) \\
{}={} & m \circ (g, f) = m_H(g, f);" title="m_H(f, g) ={} & m \circ (f \times g) = m \circ w \circ (f, g) \\
{}={} & m \circ (g, f) = m_H(g, f);">
</div>
<p>
hence, <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="18" src="https://math.fontein.de/formulae/fz.zKPyJsQRuegPik7o4JKGJGwkX9eIbtd2wHA.svgz" alt="(H, m_H)" title="(H, m_H)"></span> is abelian.
</p>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
For <span class="inline-formula"><img class="img-inline-formula img-formula" width="55" height="12" src="https://math.fontein.de/formulae/TfkE2mxa_cnslvdn73NVO7sV3O3PG.QGFfl1tA.svgz" alt="\schmC = \catSet" title="\schmC = \catSet"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="40" height="18" src="https://math.fontein.de/formulae/K2sYP9rMj7El42dxUZoIWXghBbrYW6Xl3TQq9Q.svgz" alt="(G, \cdot)" title="(G, \cdot)"></span> being a group (identified with the associated group object in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span>), we get that the group <span class="inline-formula"><img class="img-inline-formula img-formula" width="111" height="18" src="https://math.fontein.de/formulae/qMIGZA_BDzyHJq1jvnDOdqZCyiz6rcVAGM9Giw.svgz" alt="\Hom_\catSet(X, G)" title="\Hom_\catSet(X, G)"></span> defined in the proposition is exactly the set <span class="inline-formula"><img class="img-inline-formula img-formula" width="193" height="19" src="https://math.fontein.de/formulae/JxMD182n8cFvL6Jayi0wFD4_Py6npNu9VlQSaw.svgz" alt="G^X = \{ (g_i)_{x \in X} \mid g_i \in G \}" title="G^X = \{ (g_i)_{x \in X} \mid g_i \in G \}"></span>, where for elements <span class="inline-formula"><img class="img-inline-formula img-formula" width="183" height="19" src="https://math.fontein.de/formulae/gPJtQb432PpURXkPXYQ5dZ25I9SlAobbkRK1Sw.svgz" alt="(g_x)_{x\in X}, (h_x)_{x \in X} \in G^X" title="(g_x)_{x\in X}, (h_x)_{x \in X} \in G^X"></span> we define <span class="inline-formula"><img class="img-inline-formula img-formula" width="236" height="18" src="https://math.fontein.de/formulae/DTEzz6NAlQIGdl_ZRgRUlA.3NbIQx7kxf2Vbaw.svgz" alt="(g_x)_{x \in X} (h_x)_{x \in X} := (g_x h_x)_{x \in X}" title="(g_x)_{x \in X} (h_x)_{x \in X} := (g_x h_x)_{x \in X}"></span>; then <span class="inline-formula"><img class="img-inline-formula img-formula" width="159" height="22" src="https://math.fontein.de/formulae/UgRMQ6Lu4EyUm50bkU_YF4aADrCG_aHKcg3KYw.svgz" alt="(g_x)_{x \in X}^{-1} = (g_x^{-1})_{x \in X}" title="(g_x)_{x \in X}^{-1} = (g_x^{-1})_{x \in X}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="18" src="https://math.fontein.de/formulae/SwOpOwUMkGIJhMoG3BUEjEXUXgOmum4UOuL43w.svgz" alt="1_{G^X} = (1_G)_{x \in X}" title="1_{G^X} = (1_G)_{x \in X}"></span>.
</p>
<p>
In <span class="inline-formula"><img class="img-inline-formula img-formula" width="21" height="12" src="https://math.fontein.de/formulae/VOrl7F25VqQHv35P164VNh2Ckbi4H3WEeW2xrg.svgz" alt="\catSet" title="\catSet"></span>, this seems to be not too interesting, but now assume that <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> is the category of topological spaces (with continuous maps). If <span class="inline-formula"><img class="img-inline-formula img-formula" width="66" height="18" src="https://math.fontein.de/formulae/qdnS493bVA6HWXPEEDrtVi8SjkSfeDg5mBsc2g.svgz" alt="X = \{ p \}" title="X = \{ p \}"></span>, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="18" src="https://math.fontein.de/formulae/wO0ye2YjUE3eY2wtZMd4rXq5oWZvzz8_FmJj8Q.svgz" alt="\Hom_\schmC(X, G)" title="\Hom_\schmC(X, G)"></span> gives exactly the group structure of <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span>, throwing away the additional information (i.e. that the group operations are continuous with respect to the topology on <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span>). Now, if <span class="inline-formula"><img class="img-inline-formula img-formula" width="52" height="12" src="https://math.fontein.de/formulae/Irfgqb3qEZnxdhHzK6Z2tZpXDkgSB0KY.9AePQ.svgz" alt="X = \R" title="X = \R"></span> with the usual topology, the elements of <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="18" src="https://math.fontein.de/formulae/wO0ye2YjUE3eY2wtZMd4rXq5oWZvzz8_FmJj8Q.svgz" alt="\Hom_\schmC(X, G)" title="\Hom_\schmC(X, G)"></span> are continuous paths in <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span>. Hence, in <span class="inline-formula"><img class="img-inline-formula img-formula" width="98" height="18" src="https://math.fontein.de/formulae/wO0ye2YjUE3eY2wtZMd4rXq5oWZvzz8_FmJj8Q.svgz" alt="\Hom_\schmC(X, G)" title="\Hom_\schmC(X, G)"></span>, we can add continuous (parameterized) paths in <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span> (using pointwise addition)!
</p>
<p>
Now the interesting result is the following application of Yondea's lemma:
</p>
<div class="theorem-environment theorem-theorem-environment">
<div class="theorem-header theorem-theorem-header">
Theorem.
</div>
<div class="theorem-content theorem-theorem-content">
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> be a category with finite products and a final object <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="12" src="https://math.fontein.de/formulae/pYWI0gROGxbmepTlPB65gHsm8w4iqZkEdmltJA.svgz" alt="S" title="S"></span>.
</p>
<ol class="enum-level-1">
<li>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/xVBTg1pR3BXujqwBzTaHu8l8jUmSZxFPDwxJLQ.svgz" alt="(G, m, i, e)" title="(G, m, i, e)"></span> is a group object, then the assignment
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="350" height="18" src="https://math.fontein.de/formulae/R0Or1P7hoEywEEG65GoOpjxJBYNU70X7xJO4Tg.svgz" alt="\schmC \ni X \mapsto (\Hom_\schmC(X, G), (f, g) \mapsto m \circ (f \times g))" title="\schmC \ni X \mapsto (\Hom_\schmC(X, G), (f, g) \mapsto m \circ (f \times g))">
</div>
<p>
is a functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/FdgwAgeSIGP2Zh.SEW7h89WcV_eIL9gqxiJzUg.svgz" alt="\schmC \to \catGrp" title="\schmC \to \catGrp"></span>. Moreover, <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/xVBTg1pR3BXujqwBzTaHu8l8jUmSZxFPDwxJLQ.svgz" alt="(G, m, i, e)" title="(G, m, i, e)"></span> is commutative if, and only if, the image of this functor lies in the subcategory <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/3VO50a7dSjV106cpVTicUgwuuyRbMSvCH7_6Jw.svgz" alt="\catAb" title="\catAb"></span> of <span class="inline-formula"><img class="img-inline-formula img-formula" width="24" height="16" src="https://math.fontein.de/formulae/fuj0xeAJAbJYHrp_4.B.un8wkA7ZYPBfJvX47w.svgz" alt="\catGrp" title="\catGrp"></span>.
</p>
</li>
<li>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="117" height="18" src="https://math.fontein.de/formulae/3ino0gNUAwD2qcS12WSqa26gc8ElK_A_pxmhYw.svgz" alt="(G, m_G, i_G, e_G)" title="(G, m_G, i_G, e_G)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="123" height="18" src="https://math.fontein.de/formulae/JAQ3tAv1FtLS_tOT.9IWrnjB6HuH7ORfDAtY9A.svgz" alt="(H, m_H, i_H, e_H)" title="(H, m_H, i_H, e_H)"></span> be two group objects in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span>, and let <span class="inline-formula"><img class="img-inline-formula img-formula" width="133" height="16" src="https://math.fontein.de/formulae/G2ycLajA25eAEGIHj1PSS_egFexgmfafBy8tDA.svgz" alt="F_G, F_H : \schmC \to \catGrp" title="F_G, F_H : \schmC \to \catGrp"></span> be the corresponding functors. The group object homomorphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="294" height="18" src="https://math.fontein.de/formulae/BRcIXhiGIFZVtK7plEdsIC2BL8ogC6M09LgPQw.svgz" alt="\varphi : (G, m_G, i_G, e_G) \to (H, m_H, i_H, e_H)" title="\varphi : (G, m_G, i_G, e_G) \to (H, m_H, i_H, e_H)"></span> correspond one-to-one to natural transformations <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="15" src="https://math.fontein.de/formulae/_NCEodSWVJbY81.V14Dne0fWb4Vum9hEAh.H6w.svgz" alt="F_G \to F_H" title="F_G \to F_H"></span>.
</p>
<p>
In particular, two group object structures on an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span> give two naturally equivalent functors if, and only if, the group object structures are isomorphic.
</p>
</li>
<li>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="13" src="https://math.fontein.de/formulae/Vw3zO2QD64b3PmwrbOuK1vgho1W_7TCKegZoig.svgz" alt="G \in \schmC" title="G \in \schmC"></span>. There is a one-to-one correspondence between the isomorphism classes of group object structures on <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span> and natural equivalence classes of functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/FdgwAgeSIGP2Zh.SEW7h89WcV_eIL9gqxiJzUg.svgz" alt="\schmC \to \catGrp" title="\schmC \to \catGrp"></span> which are represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span>:
</p>
<p>
If <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/xVBTg1pR3BXujqwBzTaHu8l8jUmSZxFPDwxJLQ.svgz" alt="(G, m, i, e)" title="(G, m, i, e)"></span> is a group object structure, then
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="318" height="18" src="https://math.fontein.de/formulae/lQJLs3W3GTbXjTVDSw_wg2cBMfu6bJZOAhmKMA.svgz" alt="X \mapsto (\Hom_\schmC(X, G), (f, g) \mapsto m \circ (f \times g))" title="X \mapsto (\Hom_\schmC(X, G), (f, g) \mapsto m \circ (f \times g))">
</div>
<p>
is the corresponding functor.
</p>
</li>
</ol>
</div>
</div>
<p>
Before showing the theorem, note that the functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="17" src="https://math.fontein.de/formulae/3z1aFiDAKL2wSl_z1pk7IBMWfaDw4QlXtdNJgg.svgz" alt="h : \schmC \to \hat{\schmC}" title="h : \schmC \to \hat{\schmC}"></span> preserves products, i.e. if <span class="inline-formula"><img class="img-inline-formula img-formula" width="69" height="16" src="https://math.fontein.de/formulae/nn_Qx0YGNdNQ.McKxQQe0g.e4iyvVCkR.LS4Ng.svgz" alt="X, Y \in \schmC" title="X, Y \in \schmC"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="52" height="14" src="https://math.fontein.de/formulae/7UcmcHPZ4gIm75zgcmEmEYxqwrCipODoz0YO5A.svgz" alt="X \times Y" title="X \times Y"></span> exists, then <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="18" src="https://math.fontein.de/formulae/SeblBJYt_vMkDopZndgGBQL1p5sMb_Ck_L0fwA.svgz" alt="h(X) \times h(Y)" title="h(X) \times h(Y)"></span> exists in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="17" src="https://math.fontein.de/formulae/dJmVMC7OsCRBwIg5EA3R3mI6.ywENUHzo7i.oQ.svgz" alt="\hat{\schmC}" title="\hat{\schmC}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="199" height="18" src="https://math.fontein.de/formulae/70u8D6fVQaAC6GuP2WD9ZnhhQ2.9Qr436uAifA.svgz" alt="h(X \times Y) \cong h(X) \times h(Y)" title="h(X \times Y) \cong h(X) \times h(Y)"></span> in a natural way.
</p>
<div class="theorem-environment theorem-proof-environment qed">
<div class="theorem-header theorem-proof-header">
Proof.
</div>
<div class="theorem-content theorem-proof-content">
<ol class="enum-level-1">
<li>
<p>
First, we have to show that for morphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="17" src="https://math.fontein.de/formulae/cMSJpJBZ5j_mpOWNuAZ7CsgGppg8PM6j63_FGQ.svgz" alt="\varphi : X \to X'" title="\varphi : X \to X'"></span>, the induced map <span class="inline-formula"><img class="img-inline-formula img-formula" width="338" height="18" src="https://math.fontein.de/formulae/nBJZMmQmFomP2eTUQzjxM6yvSWc84xDW.0NY5Q.svgz" alt="\Hom_\schmC(\varphi, G) : \Hom_\schmC(X, G) \to \Hom_\schmC(X', G)" title="\Hom_\schmC(\varphi, G) : \Hom_\schmC(X, G) \to \Hom_\schmC(X', G)"></span> is a group homomorphism. For that, let <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="18" src="https://math.fontein.de/formulae/Yq4N6F9zKYo3ZPr01efnHtQ8vuozgTUQpQkaBQ.svgz" alt="f, g \in \Hom_\schmC(X, G)" title="f, g \in \Hom_\schmC(X, G)"></span>; then
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="355" height="71" src="https://math.fontein.de/formulae/6JVu8BPGls_YyxsaAxStbgAWAZUTBjNwLEf4kQ.svgz" alt="& \Hom_\schmC(\varphi, G)(m \circ (f \times g)) = m \circ (f \times g) \circ \varphi \\
{}={} & m \circ ((f \circ \varphi) \times (g \circ \varphi)) \\
{}={} & m \circ (\Hom_\schmC(\varphi, G)(f) \times \Hom_\schmC(\varphi, G)(g))," title="& \Hom_\schmC(\varphi, G)(m \circ (f \times g)) = m \circ (f \times g) \circ \varphi \\
{}={} & m \circ ((f \circ \varphi) \times (g \circ \varphi)) \\
{}={} & m \circ (\Hom_\schmC(\varphi, G)(f) \times \Hom_\schmC(\varphi, G)(g)),">
</div>
<p>
what we had to show.
</p>
<p>
Finally, we are left to show that the fact that the image of the functor lies in <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/3VO50a7dSjV106cpVTicUgwuuyRbMSvCH7_6Jw.svgz" alt="\catAb" title="\catAb"></span> implies that <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/xVBTg1pR3BXujqwBzTaHu8l8jUmSZxFPDwxJLQ.svgz" alt="(G, m, i, e)" title="(G, m, i, e)"></span> is commutative, i.e. that <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="8" src="https://math.fontein.de/formulae/WsKWMUu28.hceFh9sRb9QrVznyageCBrz8LNMQ.svgz" alt="m \circ w = m" title="m \circ w = m"></span>. For that, consider the natural transformations <span class="inline-formula"><img class="img-inline-formula img-formula" width="146" height="15" src="https://math.fontein.de/formulae/d35i1FekiU4Hf2ItRlB.Qz66RgCwsEmT41MfFg.svgz" alt="\hat{m} : h_G \times h_G \to h_G" title="\hat{m} : h_G \times h_G \to h_G"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="117" height="20" src="https://math.fontein.de/formulae/V1NljwEGmm1GjIw_4QswDVLONFe6ELvhGZMaig.svgz" alt="\hat{m}(X) = \cdot_{h_G(X)}" title="\hat{m}(X) = \cdot_{h_G(X)}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="187" height="15" src="https://math.fontein.de/formulae/yQYb9gmv_zUWIjFHyd_NaUSzFJAVffO92LKMlA.svgz" alt="\hat{w} : h_G \times h_G \to h_G \times h_G" title="\hat{w} : h_G \times h_G \to h_G \times h_G"></span> with <span class="inline-formula"><img class="img-inline-formula img-formula" width="149" height="18" src="https://math.fontein.de/formulae/R37_AwDrBBI3qnA6hrD4aOeB9_FQzsT3wyddaw.svgz" alt="\hat{w}(X)(x, y) = (y, x)" title="\hat{w}(X)(x, y) = (y, x)"></span>. (The fact that this is natural follows from the fact that we interpreted <span class="inline-formula"><img class="img-inline-formula img-formula" width="22" height="15" src="https://math.fontein.de/formulae/v3fdQhQFxfnfJV55lNe42C4X1bezBfbmNKkcoA.svgz" alt="h_G" title="h_G"></span> as a functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/FdgwAgeSIGP2Zh.SEW7h89WcV_eIL9gqxiJzUg.svgz" alt="\schmC \to \catGrp" title="\schmC \to \catGrp"></span>.) Then we have the commutative diagram
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="271" height="100" src="https://math.fontein.de/formulae/8y77p1Zp5gsywKJPK_oFlMpQwELNWKe.4pMdaw.svgz" alt="\xymatrix{ h_G \times h_G \ar[rr]^{\hat{w}} \ar[dr]_{\hat{m}} & & h_G \times h_G \ar[dl]^{\hat{m}} \\ & h_G & }" title="\xymatrix{ h_G \times h_G \ar[rr]^{\hat{w}} \ar[dr]_{\hat{m}} & & h_G \times h_G \ar[dl]^{\hat{m}} \\ & h_G & }">
</div>
<p>
since for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, <span class="inline-formula"><img class="img-inline-formula img-formula" width="122" height="20" src="https://math.fontein.de/formulae/zDXl7ZRQiJV7GS9u0ZFe902GzonfHPQjsvMZlg.svgz" alt="(h_G(X), \cdot_{h_G(X)})" title="(h_G(X), \cdot_{h_G(X)})"></span> is abelian. But now <span class="inline-formula"><img class="img-inline-formula img-formula" width="134" height="16" src="https://math.fontein.de/formulae/ykmDjz2axTNd6oftlTnZlV3Ml6h.4L109ezoeQ.svgz" alt="h_G \times h_G = h_{G \times G}" title="h_G \times h_G = h_{G \times G}"></span>, whence this diagram is the image of the diagram
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="230" height="92" src="https://math.fontein.de/formulae/bcN1swzc60bzkpCqfXBaqe0STNrv3Zeki.voRQ.svgz" alt="\xymatrix{ G \times G \ar[rr]^w \ar[dr]_m & & G \times G \ar[dl]^m \\ & G & }" title="\xymatrix{ G \times G \ar[rr]^w \ar[dr]_m & & G \times G \ar[dl]^m \\ & G & }">
</div>
<p>
under <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span>: obviously, <span class="inline-formula"><img class="img-inline-formula img-formula" width="74" height="18" src="https://math.fontein.de/formulae/iiVnn2mRGDyDim.7H9U2fi3w18qaZsAzQcS6oA.svgz" alt="h(w) = \hat{w}" title="h(w) = \hat{w}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="79" height="18" src="https://math.fontein.de/formulae/EvwMafCKKMy_AUrRrgab76D1KPjoKyRB.GkKkQ.svgz" alt="h(m) = \hat{m}" title="h(m) = \hat{m}"></span>. But since <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span> is faithful, it follows from <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="12" src="https://math.fontein.de/formulae/Vt_Y6a06aBiQ95c2q1ZQ2j._lkxQrw5DTTUTcQ.svgz" alt="\hat{m} \circ \hat{w} = \hat{m}" title="\hat{m} \circ \hat{w} = \hat{m}"></span> that we also have <span class="inline-formula"><img class="img-inline-formula img-formula" width="85" height="8" src="https://math.fontein.de/formulae/WsKWMUu28.hceFh9sRb9QrVznyageCBrz8LNMQ.svgz" alt="m \circ w = m" title="m \circ w = m"></span>, i.e. that <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/xVBTg1pR3BXujqwBzTaHu8l8jUmSZxFPDwxJLQ.svgz" alt="(G, m, i, e)" title="(G, m, i, e)"></span> is commutative.
</p>
</li>
<li>
<p>
By Yoneda's lemma, we get a bijection between the morphisms <span class="inline-formula"><img class="img-inline-formula img-formula" width="58" height="12" src="https://math.fontein.de/formulae/ow9zsuI.12pB1oAOtbpMnxAi4zMYCVlQInnU2Q.svgz" alt="G \to H" title="G \to H"></span> and the natural transformations <span class="inline-formula"><img class="img-inline-formula img-formula" width="218" height="16" src="https://math.fontein.de/formulae/r.NZAzSqOrEuk6zVcjJ_sjHn.uiW_ZXo9xXNxQ.svgz" alt="Forget \circ F_G \to Forget \circ F_H" title="Forget \circ F_G \to Forget \circ F_H"></span>. Hence, we have to sow that a morphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="84" height="16" src="https://math.fontein.de/formulae/0ZcSf_C.xN1ExNRQp1v2u3USVhsII3WMlgLYRA.svgz" alt="\varphi : G \to H" title="\varphi : G \to H"></span> is a homomorphism of group objects if, and only if, the corresponding natural transformation <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="17" src="https://math.fontein.de/formulae/REF06_UuXYP1MYaasnd.8tcsPEvdwCMwht2tmw.svgz" alt="h_\varphi" title="h_\varphi"></span> is actually a natural transformation <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="15" src="https://math.fontein.de/formulae/_NCEodSWVJbY81.V14Dne0fWb4Vum9hEAh.H6w.svgz" alt="F_G \to F_H" title="F_G \to F_H"></span>.
</p>
<p>
First, note that <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="11" src="https://math.fontein.de/formulae/G0SX86eTASn_9M49WF7HzDK85n7NoD6NrqM3ew.svgz" alt="\varphi" title="\varphi"></span> is a homomorphisms of group objects <span class="inline-formula"><img class="img-inline-formula img-formula" width="117" height="18" src="https://math.fontein.de/formulae/3ino0gNUAwD2qcS12WSqa26gc8ElK_A_pxmhYw.svgz" alt="(G, m_G, i_G, e_G)" title="(G, m_G, i_G, e_G)"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="123" height="18" src="https://math.fontein.de/formulae/JAQ3tAv1FtLS_tOT.9IWrnjB6HuH7ORfDAtY9A.svgz" alt="(H, m_H, i_H, e_H)" title="(H, m_H, i_H, e_H)"></span> if, and only if, <span class="inline-formula"><img class="img-inline-formula img-formula" width="184" height="18" src="https://math.fontein.de/formulae/JuvJdN8caCWK4BM1ee22elDY48_JVIgwdAs.jQ.svgz" alt="m_H \circ (\varphi \times \varphi) = \varphi \circ m_G" title="m_H \circ (\varphi \times \varphi) = \varphi \circ m_G"></span>: in case this condition holds, one can obtain <span class="inline-formula"><img class="img-inline-formula img-formula" width="118" height="15" src="https://math.fontein.de/formulae/41YOqExJ72Dgx4HI7TR1yC7IE7G1sinPwvUnAw.svgz" alt="i_H \circ \varphi = \varphi \circ i_G" title="i_H \circ \varphi = \varphi \circ i_G"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="11" src="https://math.fontein.de/formulae/Zwb1hF7EEqZ4sZOt0EUS5DzDcXXSzCSfxQ8Kfg.svgz" alt="e_H = \varphi \circ e_G" title="e_H = \varphi \circ e_G"></span> since <span class="inline-formula"><img class="img-inline-formula img-formula" width="46" height="15" src="https://math.fontein.de/formulae/k8JpY3oyv423XhQnRISrbamSve9cl9tY5QQ9Tw.svgz" alt="i_G, e_G" title="i_G, e_G"></span> are uniquely determined by <span class="inline-formula"><img class="img-inline-formula img-formula" width="27" height="10" src="https://math.fontein.de/formulae/.YW9gtUthWoRm2ZB51QLrOJc5ZSN1Bqf290TAg.svgz" alt="m_G" title="m_G"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="49" height="15" src="https://math.fontein.de/formulae/clWQso3vDAoyHU6Whe.NzmmiCpqbin_Yj_jgaw.svgz" alt="i_H, e_H" title="i_H, e_H"></span> are uniquely determined by <span class="inline-formula"><img class="img-inline-formula img-formula" width="29" height="10" src="https://math.fontein.de/formulae/onreAD4If_aM_PF44Ny04Y0joMDEfR3JuN0YtA.svgz" alt="m_H" title="m_H"></span>. (Just consider the same statement for groups: given the group operation, there is exactly one neutral element with respect to that operation and the inverses are unique as well.)
</p>
<p>
Now, since <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span> is faithful, the diagram
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="148" height="102" src="https://math.fontein.de/formulae/yy28lAqGvXHx.lVxDaFRIldw5SazZ13skN68Hw.svgz" alt="\xymatrix{ G \times G \ar[r]^{m_G} \ar[d]_{\varphi \times \varphi} & G \ar[d]^\varphi \\ H \times H \ar[r]_{m_H} & H }" title="\xymatrix{ G \times G \ar[r]^{m_G} \ar[d]_{\varphi \times \varphi} & G \ar[d]^\varphi \\ H \times H \ar[r]_{m_H} & H }">
</div>
<p>
commutes if, and only if, the diagram
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="184" height="116" src="https://math.fontein.de/formulae/i0qE5mDvHoskK_6fq_DePTYSFUU5IwUrG4E_HQ.svgz" alt="\xymatrix{ h_G \times h_G \ar[r]^{h_{m_G}} \ar[d]_{h_\varphi \times h_\varphi} & h_G \ar[d]^{h_\varphi} \\ h_H \times h_H \ar[r]_{h_{m_H}} & h_H }" title="\xymatrix{ h_G \times h_G \ar[r]^{h_{m_G}} \ar[d]_{h_\varphi \times h_\varphi} & h_G \ar[d]^{h_\varphi} \\ h_H \times h_H \ar[r]_{h_{m_H}} & h_H }">
</div>
<p>
commutes. That the first diagram commutes is equivalent to the fact that <span class="inline-formula"><img class="img-inline-formula img-formula" width="12" height="11" src="https://math.fontein.de/formulae/G0SX86eTASn_9M49WF7HzDK85n7NoD6NrqM3ew.svgz" alt="\varphi" title="\varphi"></span> is a homomorphism of group objects. That the second diagram commuts is equivalent to the fact that <span class="inline-formula"><img class="img-inline-formula img-formula" width="50" height="18" src="https://math.fontein.de/formulae/PbDxtq5svtIbQ1IREYg5WHeH0qKbZH5NJQu16w.svgz" alt="h_\varphi(X)" title="h_\varphi(X)"></span> is a group homomorphism for every <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, i.e. that <span class="inline-formula"><img class="img-inline-formula img-formula" width="20" height="17" src="https://math.fontein.de/formulae/REF06_UuXYP1MYaasnd.8tcsPEvdwCMwht2tmw.svgz" alt="h_\varphi" title="h_\varphi"></span> is a natural transformation for functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/FdgwAgeSIGP2Zh.SEW7h89WcV_eIL9gqxiJzUg.svgz" alt="\schmC \to \catGrp" title="\schmC \to \catGrp"></span>.
</p>
</li>
<li>
<p>
Part 1. shows that every group object structure on <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span> induces a functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/FdgwAgeSIGP2Zh.SEW7h89WcV_eIL9gqxiJzUg.svgz" alt="\schmC \to \catGrp" title="\schmC \to \catGrp"></span>. Part 2. shows the statement that the group structures are isomorphic if, and only if, the functors are naturally equivalent.
</p>
<p>
Let <span class="inline-formula"><img class="img-inline-formula img-formula" width="91" height="16" src="https://math.fontein.de/formulae/KIdEPCIfmExpFNfXAhMoN5hfiPGWx.aDVL8A9g.svgz" alt="F : \schmC \to \catGrp" title="F : \schmC \to \catGrp"></span> be a functor which is represented by <span class="inline-formula"><img class="img-inline-formula img-formula" width="14" height="12" src="https://math.fontein.de/formulae/1SEBa2mo2npdO9VAigPQfAPRoqEDUsQLVvcYYg.svgz" alt="G" title="G"></span>, i.e. there exists a natural equivalence <span class="inline-formula"><img class="img-inline-formula img-formula" width="233" height="18" src="https://math.fontein.de/formulae/J7xPP2Zh7eECwh4Kaup6UZyvADanr0ZPt_G_Dg.svgz" alt="\eta : \Hom_\schmC(-, G) \to Forget \circ F" title="\eta : \Hom_\schmC(-, G) \to Forget \circ F"></span>. For <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, define
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="383" height="48" src="https://math.fontein.de/formulae/7Uhb6PL2KW27G7nnrB0Vvd.emc8SnOzA6e3cxg.svgz" alt="\hat{m}_X :{} & \Hom_\schmC(A, G) \times \Hom_\schmC(A, G) \to \Hom_\schmC(A, G) \\
& (f, g) \mapsto \eta(A)^{-1}(\eta(A)(f) \cdot_{F(A)} \eta(A)(f))." title="\hat{m}_X :{} & \Hom_\schmC(A, G) \times \Hom_\schmC(A, G) \to \Hom_\schmC(A, G) \\
& (f, g) \mapsto \eta(A)^{-1}(\eta(A)(f) \cdot_{F(A)} \eta(A)(f)).">
</div>
<p>
As <span class="inline-formula"><img class="img-inline-formula img-formula" width="9" height="11" src="https://math.fontein.de/formulae/LJX59haeTalq4bSnxWApTe0lMbaooEzjD2TZqQ.svgz" alt="\eta" title="\eta"></span> is a natural transformation, it turns out that <span class="inline-formula"><img class="img-inline-formula img-formula" width="100" height="15" src="https://math.fontein.de/formulae/U0yd7tlqHFW9V2Mf24KIGfrA89ZTQYzYjm2DfQ.svgz" alt="\hat{m} : A \mapsto \hat{m}_X" title="\hat{m} : A \mapsto \hat{m}_X"></span> gives a natural transformation
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="478" height="18" src="https://math.fontein.de/formulae/UJxsJ1SSpPqvgnRBTFQlyktAoeeUb04t_NMlOQ.svgz" alt="h_G \times h_G = \Hom_\schmC(-, G) \times \Hom_\schmC(-, G) \to \Hom_\schmC(-, G) = h_G." title="h_G \times h_G = \Hom_\schmC(-, G) \times \Hom_\schmC(-, G) \to \Hom_\schmC(-, G) = h_G.">
</div>
<p>
Since <span class="inline-formula"><img class="img-inline-formula img-formula" width="134" height="16" src="https://math.fontein.de/formulae/JVMQe20bvUyhMVJ2x5tHjwlmEaoH1dibRigkQw.svgz" alt="h_G \times h_G \cong h_{G \times G}" title="h_G \times h_G \cong h_{G \times G}"></span> in a natural way, we have that <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/AkDAaxQe1werKtQ6StJO2EH3m5Fo1_b8Fn8XFw.svgz" alt="\hat{m}" title="\hat{m}"></span> gives a natural transformation <span class="inline-formula"><img class="img-inline-formula img-formula" width="94" height="16" src="https://math.fontein.de/formulae/228gZupbfZ6UV0wRzzbzDx22gp6yA_uUMCuwrg.svgz" alt="h_{G \times G} \to h_G" title="h_{G \times G} \to h_G"></span>. By the <a href="https://math.fontein.de/2009/08/16/fun-with-representable-functors-or-why-i-like-yondeas-lemma/#yoneda-corollary">corollary</a> on Yoneda's lemma, this natural transformation corresponds to a morphism <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="14" src="https://math.fontein.de/formulae/NMqsbAaH46PqOWuFBUwRZmOEPxV441fKQyRRww.svgz" alt="m : G \times G \to G" title="m : G \times G \to G"></span>.
</p>
<p>
Let us show that <span class="inline-formula"><img class="img-inline-formula img-formula" width="121" height="14" src="https://math.fontein.de/formulae/NMqsbAaH46PqOWuFBUwRZmOEPxV441fKQyRRww.svgz" alt="m : G \times G \to G" title="m : G \times G \to G"></span> satisfies the associativity diagram, i.e. we have that
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="295" height="104" src="https://math.fontein.de/formulae/.ztIPSfrfmkZzWQkhXA19DE3vyaO0l88P_T_cQ.svgz" alt="\xymatrix@C+1cm{ G \times G \times G \ar[r]^{m \times \id_G} \ar[d]_{\qquad \id_G \times m} & G \times G \ar[d]^m \\ G \times G \ar[r]_m & G }" title="\xymatrix@C+1cm{ G \times G \times G \ar[r]^{m \times \id_G} \ar[d]_{\qquad \id_G \times m} & G \times G \ar[d]^m \\ G \times G \ar[r]_m & G }">
</div>
<p>
commutes.
</p>
<p>
Consider the natural transformations
</p>
<div class="align-formula">
<img class="img-align-formula img-formula" width="332" height="44" src="https://math.fontein.de/formulae/GIBqKcEf62eR7ukGyJ_PxEL0DhOhBg9qNK_VNQ.svgz" alt="\hat{m} \circ \id :{} & (h_G \times h_G) \to h_G \to h_G \times h_G \\
\text{and} \quad \id \times \hat{m} :{} & h_G \times (h_G \times h_G) \to h_G \times h_G." title="\hat{m} \circ \id :{} & (h_G \times h_G) \to h_G \to h_G \times h_G \\
\text{and} \quad \id \times \hat{m} :{} & h_G \times (h_G \times h_G) \to h_G \times h_G.">
</div>
<p>
From the definition of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/AkDAaxQe1werKtQ6StJO2EH3m5Fo1_b8Fn8XFw.svgz" alt="\hat{m}" title="\hat{m}"></span> it follows that the following diagram commutes:
</p>
<div class="display-formula">
<img class="img-display-formula img-formula" width="239" height="108" src="https://math.fontein.de/formulae/Ge5OfEO1kpfu_ZKKUWza7SIioslyxOzOOrVu6Q.svgz" alt="\xymatrix{ h_G \times h_G \times h_G \ar[r]^{\hat{m} \times \id} \ar[d]_{\id \times \hat{m}} & h_G \times h_G \ar[d]^{\hat{m}} \\ h_G \times h_G \ar[r]_{\hat{m}} & h_G }" title="\xymatrix{ h_G \times h_G \times h_G \ar[r]^{\hat{m} \times \id} \ar[d]_{\id \times \hat{m}} & h_G \times h_G \ar[d]^{\hat{m}} \\ h_G \times h_G \ar[r]_{\hat{m}} & h_G }">
</div>
<p>
(Simply plug in an object <span class="inline-formula"><img class="img-inline-formula img-formula" width="48" height="13" src="https://math.fontein.de/formulae/WuIBEO3kV90Yp4VzpGxWqUZUmNir7jptZfG5.w.svgz" alt="X \in \schmC" title="X \in \schmC"></span>, and then use the definition of <span class="inline-formula"><img class="img-inline-formula img-formula" width="16" height="12" src="https://math.fontein.de/formulae/AkDAaxQe1werKtQ6StJO2EH3m5Fo1_b8Fn8XFw.svgz" alt="\hat{m}" title="\hat{m}"></span> and the fact that <span class="inline-formula"><img class="img-inline-formula img-formula" width="106" height="20" src="https://math.fontein.de/formulae/OwaxDjoBgIOOu5fv6zDyDkPbgcJS5kyCjvMaYg.svgz" alt="(F(X), \cdot_{F(X)})" title="(F(X), \cdot_{F(X)})"></span> is a group.) Now, since <span class="inline-formula"><img class="img-inline-formula img-formula" width="134" height="16" src="https://math.fontein.de/formulae/ykmDjz2axTNd6oftlTnZlV3Ml6h.4L109ezoeQ.svgz" alt="h_G \times h_G = h_{G \times G}" title="h_G \times h_G = h_{G \times G}"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="200" height="16" src="https://math.fontein.de/formulae/uEGvBAipTjsB.Fq9PU9wK21kAXywBrmv0JrQzQ.svgz" alt="h_G \times h_G \times h_G = h_{G \times G \times G}" title="h_G \times h_G \times h_G = h_{G \times G \times G}"></span>, one can see that this diagram is the image of the previous diagram under <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span>. But since <span class="inline-formula"><img class="img-inline-formula img-formula" width="10" height="12" src="https://math.fontein.de/formulae/BhIgQy.yStNYm82Tu7qgDRTe5Zg2.7Iux8hXFw.svgz" alt="h" title="h"></span> is faithful, the previous diagram also has to commute.
</p>
<p>
Now, one can obtain <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="12" src="https://math.fontein.de/formulae/sOaXQ02qOl_MkP6vwTXZqaJXjDYqRF5qOWBnwg.svgz" alt="i : G \to G" title="i : G \to G"></span> and <span class="inline-formula"><img class="img-inline-formula img-formula" width="76" height="12" src="https://math.fontein.de/formulae/eQjeUCX4ZxzafGojzGu5E4_MePxVGfUVb14jRw.svgz" alt="e : S \to G" title="e : S \to G"></span> in the same manner and show that they satisfy the conditions they have to such that <span class="inline-formula"><img class="img-inline-formula img-formula" width="81" height="18" src="https://math.fontein.de/formulae/xVBTg1pR3BXujqwBzTaHu8l8jUmSZxFPDwxJLQ.svgz" alt="(G, m, i, e)" title="(G, m, i, e)"></span> is a group object.
</p>
</li>
</ol>
</div>
<div class="qed-block"><span class="qed-sign"></span></div>
</div>
<p>
In a nutshell, this result says that representable functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/FdgwAgeSIGP2Zh.SEW7h89WcV_eIL9gqxiJzUg.svgz" alt="\schmC \to \catGrp" title="\schmC \to \catGrp"></span> are the same as group objects in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span>, and that representable functors <span class="inline-formula"><img class="img-inline-formula img-formula" width="57" height="12" src="https://math.fontein.de/formulae/d0kRJFIuUkrEBMrZU5NOsY_5v.0qLzp3iHyrpQ.svgz" alt="\schmC \to \catAb" title="\schmC \to \catAb"></span> are the same as commutative group objects in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span>. This is somewhat surprising, as one expects that constructing a group object structure on an object in <span class="inline-formula"><img class="img-inline-formula img-formula" width="11" height="12" src="https://math.fontein.de/formulae/.amQtip_2bB7MHc5cDQcOMYB6QKCK2XH8XVZzA.svgz" alt="\schmC" title="\schmC"></span> is hard, while coming up with a (representable) functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/FdgwAgeSIGP2Zh.SEW7h89WcV_eIL9gqxiJzUg.svgz" alt="\schmC \to \catGrp" title="\schmC \to \catGrp"></span> (or <span class="inline-formula"><img class="img-inline-formula img-formula" width="19" height="12" src="https://math.fontein.de/formulae/3VO50a7dSjV106cpVTicUgwuuyRbMSvCH7_6Jw.svgz" alt="\catAb" title="\catAb"></span>) sounds easier. In fact, constructing a functor <span class="inline-formula"><img class="img-inline-formula img-formula" width="62" height="16" src="https://math.fontein.de/formulae/FdgwAgeSIGP2Zh.SEW7h89WcV_eIL9gqxiJzUg.svgz" alt="\schmC \to \catGrp" title="\schmC \to \catGrp"></span> is usually easier, but showing that the constructed functor is representable is hard. (Just consider the construction of <a href="https://en.wikipedia.org/wiki/Picard_scheme">Picard schemes</a> or <a href="https://en.wikipedia.org/wiki/Hilbert_scheme">Hilbert schemes</a>.)
</p>
<p>
If you are interested in literature, see, for example, the book “Néron Models” by S. Bosch, W. Lütkebohmert and M. Raynaud (Ergebnisse der Mathematik und ihrer Grenzgebiete no. 21, Springer, 1990), and the book “Commutative Group Schemes” by F. Oort (Lecture Notes in Mathematics no. 15, Springer, 1966).
</p>
</div>