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  "body": "\n## Summary\n\nA contravariant functor is an assignment like a [[functor|28NQ0KRHD3]] that reverses the direction of [[morphisms|859F0TFVA3]]: it sends $f \\colon A \\to B$ to a morphism $F(B) \\to F(A)$.\nIt is the same thing as an ordinary functor out of the [[opposite category|DAMK8XE1FR]].\nThe [[dual space|07JBSZE0V4]] construction, cohomology and [[presheaves|AND2WHZC8P]] are contravariant functors.\nThe hom functor $\\Hom(-, X)$ is the basic example, and the Yoneda lemma is about such functors.\n\n## Explanations\n\n### L1 human\n\nSome constructions turn arrows around.\nA map of sets $f \\colon A \\to B$ pulls back each subset of $B$ to a subset of $A$, its preimage.\nSo \"subsets of\" sends $f$ to a map going backwards, from subsets of $B$ to subsets of $A$.\nSuch a backward rule that still respects identities and composition is a contravariant functor.\nComposition flips order too: pulling back along $g \\circ f$ is pulling back along $g$, then along $f$.\n",
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