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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]], let $\\mathcal{C}^{\\mathrm{op}}$ be its [[opposite category|DAMK8XE1FR]], and let $\\mathbf{Set}$ be the category of sets and functions.\nA *presheaf* on $\\mathcal{C}$ is a [[functor|28NQ0KRHD3]] $F \\colon \\mathcal{C}^{\\mathrm{op}} \\to \\mathbf{Set}$.\nSo $F$ assigns a set $F(X)$ to each object $X$ and a function $F(f) \\colon F(Y) \\to F(X)$ to each morphism $f \\colon X \\to Y$, with $F(\\id_X) = \\id_{F(X)}$ and $F(g \\circ f) = F(f) \\circ F(g)$.\nA morphism of presheaves is a [[natural transformation|KYCPCC57FT]], and the presheaves on $\\mathcal{C}$ form the category $[\\mathcal{C}^{\\mathrm{op}}, \\mathbf{Set}]$, often written $\\widehat{\\mathcal{C}}$.\n\n## Remarks\n\nExample: let $T$ be a [[topological space|0CQHFPGJ45]] and let $\\mathcal{O}(T)$ be the category whose objects are the [[open sets|02BH3MCBQB]] of $T$, with one morphism $V \\to U$ when $V \\subseteq U$.\nLet $F(U)$ be the set of continuous functions $U \\to \\R$ and let $F(V \\to U)$ be restriction $s \\mapsto s|_V$.\nRestriction to $U$ is the identity and restricting to $V$ then to $W$ equals restricting to $W$, so $F$ is a presheaf.\n\nNon-example: on $\\mathcal{O}(T)$, let $G(U)$ be the set of all functions $U \\to \\R$ and send $V \\subseteq U$ to extension by zero $G(V) \\to G(U)$.\nThis is a functor $\\mathcal{O}(T) \\to \\mathbf{Set}$, but its maps go from $V$ to $U$, so it is covariant and not a presheaf on $\\mathcal{O}(T)$.\n\nEvery hom functor $\\Hom(-, A)$ is a presheaf, and a presheaf on $\\mathcal{O}(T)$ is a presheaf on $T$ in the sense of sheaf theory.\n",
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