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  "body": "\n## Statement\n\nLet $\\mathcal{C}$ be a locally small [[category|A2DE66937D]] and let $A$ be an object of $\\mathcal{C}$.\nLet $\\mathbf{Set}$ be the category of sets and let $\\mathcal{C}^{\\mathrm{op}}$ be the [[opposite category|DAMK8XE1FR]].\nLet $F \\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathbf{Set}$ be a [[presheaf|AND2WHZC8P]] on $\\mathcal{C}$.\nLet $\\mathrm{Nat}(\\Hom(-, A), F)$ be the class of [[natural transformations|KYCPCC57FT]] from the [[hom functor|VJXS8AK67E]] $\\Hom(-, A)$ to $F$.\nThen the map\n$$\\mathrm{Nat}(\\Hom(-, A), F) \\to F(A), \\qquad \\alpha \\mapsto \\alpha_A(\\id_A)$$\nis a bijection.\nIts inverse sends $x \\in F(A)$ to the natural transformation $\\alpha$ with $\\alpha_X(f) = F(f)(x)$ for every object $X$ and every $f \\colon X \\to A$.\nThe bijection is natural in both $A$ and $F$.\n\n## Remarks\n\nProof idea: let $\\alpha$ be natural and $x = \\alpha_A(\\id_A)$.\nFor $f \\colon X \\to A$, naturality applied to $f$ and the element $\\id_A$ gives $\\alpha_X(\\id_A \\circ f) = F(f)(\\alpha_A(\\id_A))$, that is, $\\alpha_X(f) = F(f)(x)$.\nSo $\\alpha$ is determined by $x$, and a direct check shows that every $x$ defines a natural transformation in this way.\n\nExample: take $F = \\Hom(-, B)$ for an object $B$.\nThe lemma says that natural transformations $\\Hom(-, A) \\to \\Hom(-, B)$ correspond to elements of $\\Hom(A, B)$, that is, to morphisms $A \\to B$; each one is post-composition with that morphism.\n\nThe covariant form is the same: for a functor $G \\colon \\mathcal{C} \\to \\mathbf{Set}$, natural transformations $\\Hom(A, -) \\to G$ correspond to elements of $G(A)$.\nLocal smallness is needed for $\\Hom(-, A)$ to take values in $\\mathbf{Set}$.\n\n## Explanations\n\n### L1 human\n\nThe Yoneda lemma says that an object is known completely by the arrows that point into it.\nA natural way of turning arrows into $A$ into elements of some set-valued structure $F$ is decided by one choice: where the identity arrow of $A$ goes.\nEverything else is forced by naturality.\n\n### L2 human\n\nThe lemma is the reason that the Yoneda embedding $A \\mapsto \\Hom(-, A)$ is fully faithful, so every category sits inside a category of presheaves without loss of information.\nIt implies that a representing object of a representable functor is unique up to unique isomorphism.\nThis is why objects defined by a universal property are unique up to unique isomorphism.\nIn practice it is used to prove that two objects are isomorphic by showing that their hom functors are naturally isomorphic.\nApplied to a group seen as a one-object category, it gives Cayley's theorem, that every group embeds in a group of permutations of its own elements.\n",
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  "name": "Yoneda lemma",
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