Definition:Covariant Hom Functor

Definition
Let $\mathbf{Set}$ be the category of sets.

Let $\mathbf C$ be a locally small category. Let $C \in \mathbf C_0$ be an object of $\mathbf C$.

The covariant hom functor based at $C$, $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right): \mathbf C \to \mathbf{Set}$, is the covariant functor defined by:

where $\operatorname{Hom}_{\mathbf C} \left({C, D}\right)$ denotes a hom set.

Thus, the morphism functor is defined to be postcomposition.

That $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right)$ is a functor is shown on Covariant Hom Functor is Functor.

Also known as
Some sources call a hom functor a representable functor.

Also denoted as
All notations for hom classes can be seen for hom functors too.

A covariant hom functor can also be denoted $h^x$; see the Yoneda embedding.

Also see

 * Contravariant Hom Functor