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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a locally small [[category|A2DE66937D]], so that for all objects $X, Y$ the [[morphisms|859F0TFVA3]] $X \\to Y$ form a [[set|0BSJ7RQ9BW]] $\\Hom(X, Y)$.\nLet $\\mathbf{Set}$ be the category of sets and functions, and let $A$ be an object of $\\mathcal{C}$.\n\n1. The *covariant hom functor* $\\Hom(A, -) \\colon \\mathcal{C} \\to \\mathbf{Set}$ sends an object $X$ to $\\Hom(A, X)$ and a morphism $f \\colon X \\to Y$ to the function $\\Hom(A, f) \\colon \\Hom(A, X) \\to \\Hom(A, Y)$, $g \\mapsto f \\circ g$.\n2. The *contravariant hom functor* $\\Hom(-, A) \\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathbf{Set}$ sends an object $X$ to $\\Hom(X, A)$ and a morphism $f \\colon X \\to Y$ to the function $\\Hom(f, A) \\colon \\Hom(Y, A) \\to \\Hom(X, A)$, $h \\mapsto h \\circ f$.\n\n## Remarks\n\nAssociativity of composition gives $\\Hom(A, f' \\circ f) = \\Hom(A, f') \\circ \\Hom(A, f)$, and $\\Hom(A, \\id_X)$ is the identity, so clause 1 defines a [[functor|28NQ0KRHD3]].\nClause 2 defines a contravariant functor, that is, a functor on the [[opposite category|DAMK8XE1FR]] $\\mathcal{C}^{\\mathrm{op}}$.\nThe two together form the bifunctor $\\Hom(-, -) \\colon \\mathcal{C}^{\\mathrm{op}} \\times \\mathcal{C} \\to \\mathbf{Set}$ on the [[product category|AFNE9C4NH6]].\nOther notations are $\\mathcal{C}(A, X)$, $h^A = \\Hom(A, -)$ and $h_A = \\Hom(-, A)$.\n\nExample: in the category of real vector spaces, take $A = \\R$.\nA linear map $\\R \\to V$ is determined by the image of $1$, so $\\Hom(\\R, V)$ is in bijection with $V$, and $\\Hom(\\R, f)$ corresponds to $f$ itself.\n\nNon-example: the assignment $X \\mapsto \\Hom(A, X)$ with $f \\mapsto (g \\mapsto g \\circ f)$ is not a covariant hom functor.\nFor $f \\colon X \\to Y$ and $g \\colon A \\to X$, the composite $g \\circ f$ is in general not defined.\nSo it does not give a function $\\Hom(A, X) \\to \\Hom(A, Y)$.\n",
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