# 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:

Object functor: | \(\ds \operatorname{Hom}_{\mathbf C} \left({C, D}\right) = \operatorname{Hom}_{\mathbf C} \left({C, D}\right) \) | |||||||

Morphism functor: | \(\ds \operatorname{Hom}_{\mathbf C} \left({C, f}\right): \operatorname{Hom}_{\mathbf C} \left({C, A}\right) \to \operatorname{Hom}_{\mathbf C} \left({C, B}\right), g \mapsto f \circ g \) | for $f: A \to B$ |

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

## Sources

- 2010: Steve Awodey:
*Category Theory*(2nd ed.) ... (previous) ... (next): $\S 2.7$