Definition:Covariant Hom Functor

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

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

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

Thus, the morphism functor is defined to be postcomposition.

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

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 covariant hom functor can also be denoted $h^x$; see the Yoneda embedding.

Also see