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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a locally small [[category|A2DE66937D]], and let $[\\mathcal{C}^{\\mathrm{op}}, \\mathbf{Set}]$ be the [[functor category|1X0WKFQX8K]] of [[presheaves|AND2WHZC8P]] on $\\mathcal{C}$, with [[natural transformations|KYCPCC57FT]] as morphisms.\nThe *Yoneda embedding* is the [[functor|28NQ0KRHD3]] $y \\colon \\mathcal{C} \\to [\\mathcal{C}^{\\mathrm{op}}, \\mathbf{Set}]$ given by:\n\n1. on objects, $y(A) = \\Hom(-, A)$, the contravariant [[hom functor|VJXS8AK67E]] of $A$;\n2. on morphisms, for $f \\colon A \\to B$, $y(f) \\colon \\Hom(-, A) \\to \\Hom(-, B)$ is the natural transformation with components $y(f)_X(g) = f \\circ g$ for $g \\colon X \\to A$.\n\n## Remarks\n\nThe components $y(f)_X$ are natural in $X$ because $f \\circ (g \\circ h) = (f \\circ g) \\circ h$.\nBy the [[Yoneda lemma|D6DF1ARPVD]] with $F = \\Hom(-, B)$, the map $f \\mapsto y(f)$ is a bijection $\\Hom(A, B) \\to \\mathrm{Nat}(y(A), y(B))$, so $y$ is [[full and faithful|GRG8WEBSH5]].\nSince $y$ is fully faithful, $y(A) \\cong y(B)$ implies $A \\cong B$.\nThe dual functor $A \\mapsto \\Hom(A, -)$ embeds $\\mathcal{C}^{\\mathrm{op}}$ into $[\\mathcal{C}, \\mathbf{Set}]$.\n\nExample: let $P$ be a preorder seen as a category, with one morphism $a \\to b$ when $a \\le b$.\nThen $y(a)$ is the presheaf with $y(a)(x)$ a one-point set when $x \\le a$ and empty otherwise, so $y$ sends $a$ to the down-set $\\set{x : x \\le a}$.\nThe map $a \\mapsto \\set{x : x \\le a}$ is the classical embedding of a preorder into its down-sets ordered by inclusion.\n\nNon-example: the assignment $A \\mapsto \\Hom(A, -)$ is not a functor $\\mathcal{C} \\to [\\mathcal{C}, \\mathbf{Set}]$; a morphism $f \\colon A \\to B$ induces $\\Hom(B, -) \\to \\Hom(A, -)$, which goes the wrong way, so clause 2 fails for this assignment.\n",
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