# Definition:Contravariant Hom Functor

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## 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: | \(\ds \operatorname{Hom}_{\mathbf C} \left({B, C}\right) = \operatorname{Hom}_{\mathbf C} \left({B, C}\right) \) | ||||||||

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

This article is complete as far as it goes, but it could do with expansion.In particular: And this is inconvenient because of concept overloading, but we haven't covered the other type of representable functorYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Expand}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Also denoted as

All notations for hom classes can be seen for **hom functors** too.

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: The above should be explicatedYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

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*(2nd ed.) ... (previous) ... (next): $\S 5.5$