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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a locally small [[category|A2DE66937D]] and let $\\mathbf{Set}$ be the category of [[sets|0BSJ7RQ9BW]].\nA [[functor|28NQ0KRHD3]] $F \\colon \\mathcal{C} \\to \\mathbf{Set}$ is *representable* if there are an object $A$ of $\\mathcal{C}$ and a natural isomorphism $\\Hom(A, -) \\cong F$.\nA natural isomorphism is a [[natural transformation|KYCPCC57FT]] whose components are all bijections.\nA functor $G \\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathbf{Set}$ is *representable* if there are an object $A$ and a natural isomorphism $\\Hom(-, A) \\cong G$.\nHere $\\Hom(A, -)$ and $\\Hom(-, A)$ are the [[hom functors|VJXS8AK67E]], and $A$ is called a *representing object*.\n\n## Remarks\n\nA representation $\\Phi \\colon \\Hom(A, -) \\cong F$ is determined by the element $u = \\Phi_A(\\id_A) \\in F(A)$, called a universal element.\nThe pair $(A, u)$ represents $F$ exactly when for every object $X$ and every $x \\in F(X)$ there is a unique $f \\colon A \\to X$ with $F(f)(u) = x$.\n\nExample: the contravariant [[power set|00E0YZCERH]] functor $P \\colon \\mathbf{Set}^{\\mathrm{op}} \\to \\mathbf{Set}$ sends $X$ to its set of subsets and $f \\colon X \\to Y$ to the preimage map $S \\mapsto f^{-1}(S)$.\nIt is represented by $A = \\{0, 1\\}$: a function $\\chi \\colon X \\to \\{0, 1\\}$ corresponds to the subset $\\chi^{-1}(1)$, and this bijection commutes with preimages.\n\nNon-example: the constant functor $F \\colon \\mathbf{Set} \\to \\mathbf{Set}$ with value the empty set is not representable.\nFor every set $A$ the set $\\Hom(A, A)$ contains $\\id_A$, so it is not in bijection with $F(A) = \\emptyset$.\n",
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