# Definition:Contravariant 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 **contravariant hom functor based at $C$**:

- $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right): \mathbf C \to \mathbf{Set}$

is the covariant functor defined by:

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

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

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

Thus, the morphism functor is defined to be precomposition.

## 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 **contravariant hom functor** can also be denoted $h_x$; see the Yoneda embedding.

## Also see

- Contravariant Hom Functor is Functor, where it is shown that $\operatorname{Hom}_{\mathbf C} \left({\cdot, C}\right)$ is a functor.

## Sources

- 2010: Steve Awodey:
*Category Theory*... (previous) ... (next): $\S 5.5$