Definition:Contravariant Representable 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 representable 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.

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

Also see

 * Covariant Representable Functor