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  "body": "\n## Definition\n\nLet $F \\colon \\mathcal{C} \\to \\mathcal{D}$ be [[left adjoint|TZHF6ZM92S]] to $G \\colon \\mathcal{D} \\to \\mathcal{C}$ by the natural bijections $\\varphi_{c, d} \\colon \\Hom_{\\mathcal{D}}(F c, d) \\to \\Hom_{\\mathcal{C}}(c, G d)$.\n\n1. The *unit* of the adjunction is the family $\\eta_c = \\varphi_{c, F c}(\\id_{F c}) \\colon c \\to G F c$ for objects $c$ of $\\mathcal{C}$.\n2. The *counit* of the adjunction is the family $\\varepsilon_d = \\varphi_{G d, d}^{-1}(\\id_{G d}) \\colon F G d \\to d$ for objects $d$ of $\\mathcal{D}$.\n\n## Remarks\n\nBoth families are [[natural transformations|KYCPCC57FT]], $\\eta \\colon \\mathrm{Id}_{\\mathcal{C}} \\Rightarrow G F$ and $\\varepsilon \\colon F G \\Rightarrow \\mathrm{Id}_{\\mathcal{D}}$.\nThey satisfy the *triangle identities*\n$$\\varepsilon_{F c} \\circ F(\\eta_c) = \\id_{F c}, \\qquad G(\\varepsilon_d) \\circ \\eta_{G d} = \\id_{G d}.$$\nThey recover $\\varphi$ by $\\varphi(g) = G(g) \\circ \\eta_c$ for $g \\colon F c \\to d$.\n\nExample: for the free vector space functor $F$ over a field $k$ and the forgetful functor $G$, the unit $\\eta_X \\colon X \\to G F X$ sends $x$ to the basis vector $e_x$.\nThe counit $\\varepsilon_V \\colon F G V \\to V$ sends the basis vector $e_v$ to the vector $v$.\nThen $\\varepsilon_{F X}(F(\\eta_X)(e_x)) = \\varepsilon_{F X}(e_{e_x}) = e_x$, which checks the first triangle identity.\n\nNon-example: if the characteristic of $k$ is not $2$, the natural maps $\\eta'_X(x) = 2 e_x$ are not a unit for this counit.\nHere $\\varepsilon_{F X} \\circ F(\\eta'_X)$ is multiplication by $2$, so the first triangle identity fails.\n",
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