Definition:Contravariant Hom Functor

Definition
Let $\mathbf C$ be a locally small category. Let $x \in C$ be an object.

The contravariant hom-functor of $x$ is the contravariant functor $\operatorname{Hom}(-, x) : \mathbf C \to \mathbf {Set}$ to the category of sets with:
 * $\operatorname{Hom}(a, x)$ is the hom class
 * If $f : a \to b$ is a morphism, $\operatorname{Hom}(f, x) : \operatorname{Hom}(b, x) \to \operatorname{Hom}(a, x)$ is the precomposition with $f$

Also denoted as
All notations for hom classes can be seen for hom functors too. It can also be denoted $h_x$; see the Yoneda embedding.

Also see

 * Definition:Covariant Hom Functor