Definition:Covariant 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 covariant representable functor based at $C$ $\operatorname{Hom}_{\mathbf C} \left({C, \cdot}\right): \mathbf C \to \mathbf{Set}$ is the covariant functor defined by:

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

Thus, the morphism functor is defined to be precomposition.

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