Contravariant functor

Summary

A contravariant functor is an assignment like a functor that reverses the direction of morphisms: it sends $f \colon A \to B$ to a morphism $F(B) \to F(A)$. It is the same thing as an ordinary functor out of the opposite category. The dual space construction, cohomology and presheaves are contravariant functors. The hom functor $\Hom(-, X)$ is the basic example, and the Yoneda lemma is about such functors.