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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ and $\\mathcal{D}$ be categories.\nA *contravariant functor* $F$ from $\\mathcal{C}$ to $\\mathcal{D}$ consists of:\n\n1. for each object $A$ of $\\mathcal{C}$, an object $F(A)$ of $\\mathcal{D}$;\n2. for each morphism $f \\colon A \\to B$ of $\\mathcal{C}$, a morphism $F(f) \\colon F(B) \\to F(A)$ of $\\mathcal{D}$;\n\nsuch that:\n\n1. $F(1_A) = 1_{F(A)}$ for every object $A$;\n2. $F(g \\circ f) = F(f) \\circ F(g)$ for all $f \\colon A \\to B$ and $g \\colon B \\to C$.\n\n## Remarks\n\nA contravariant functor from $\\mathcal{C}$ to $\\mathcal{D}$ is the same as a [[functor|28NQ0KRHD3]] $\\mathcal{C}^{\\mathrm{op}} \\to \\mathcal{D}$ from the [[opposite category|DAMK8XE1FR]], via $f^{\\mathrm{op}} \\mapsto F(f)$.\nExample: the [[power set|00E0YZCERH]] with preimages, $A \\mapsto \\mathcal{P}(A)$ and $f \\mapsto f^{-1}$, is a contravariant functor from sets to sets.\nHere $f^{-1} \\colon \\mathcal{P}(B) \\to \\mathcal{P}(A)$, and $(g \\circ f)^{-1}(S) = f^{-1}(g^{-1}(S))$ gives clause 2.\nAnother example is the [[dual space|07JBSZE0V4]] functor $V \\mapsto V^*$, $f \\mapsto f^*$ on vector spaces over a field, where $(g \\circ f)^* = f^* \\circ g^*$.\nNon-example: the power set with direct images, $f \\mapsto (S \\mapsto f(S))$, is not contravariant, because $f(\\cdot)$ goes from $\\mathcal{P}(A)$ to $\\mathcal{P}(B)$, so clause 2 of the data fails; it is a covariant functor.\n",
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