Definition:Covariant Hom Functor

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

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