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:

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

 * Definition:Covariant Hom Functor


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