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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be [[categories|A2DE66937D]], and let $F \\colon \\mathcal{C} \\to \\mathcal{D}$ and $G \\colon \\mathcal{D} \\to \\mathcal{C}$ be [[functors|28NQ0KRHD3]].\nAn *adjunction* between $F$ and $G$ is a family of bijections\n$$\\varphi_{c, d} \\colon \\Hom_{\\mathcal{D}}(F c, d) \\to \\Hom_{\\mathcal{C}}(c, G d),$$\none for each object $c$ of $\\mathcal{C}$ and each object $d$ of $\\mathcal{D}$.\nThe family must be natural in $c$ and in $d$.\nNaturality means $\\varphi(k \\circ g \\circ F h) = G k \\circ \\varphi(g) \\circ h$.\nThis must hold for all $h \\colon c' \\to c$, $g \\colon F c \\to d$ and $k \\colon d \\to d'$.\nIf an adjunction exists, $F$ is *left adjoint* to $G$ and $G$ is *right adjoint* to $F$, written $F \\dashv G$.\n\n## Remarks\n\nEquivalently, $\\varphi$ is a [[natural isomorphism|KYCPCC57FT]] between the two [[hom functors|VJXS8AK67E]] $\\Hom_{\\mathcal{D}}(F -, -)$ and $\\Hom_{\\mathcal{C}}(-, G -)$ from $\\mathcal{C}^{\\mathrm{op}} \\times \\mathcal{D}$ to sets.\n\nExample: let $F$ send a set $X$ to the free group $F(X)$, and let $G$ send a group to its underlying set.\nA homomorphism $g \\colon F(X) \\to H$ is determined by its values on the generators, and these values can be any elements of $H$.\nSo $\\varphi(g) = g|_X$ is a bijection, and it is natural in $X$ and $H$.\n\nNon-example: the forgetful functor $G$ from groups to sets has no right adjoint.\nIndeed, a left adjoint preserves colimits, hence the initial object.\nBut $G$ sends the trivial group, which is initial, to a one-point set, which is not initial in sets.\n\nAnother standard example is $- \\times Y \\dashv (-)^Y$ in sets, where $Z^Y$ is the set of maps $Y \\to Z$.\n",
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