Covariant Hom functor

Definition

Suppose $\mathcal{C}$ is a category and $A\in \mathrm{O}\mathrm{b}\mathcal{C}$. The covariant Hom functor corresponding to $A$ is defined as a covariant functor from $\mathcal{C}$ to the category of sets given as follows:

• On objects: An object $B\in \mathrm{O}\mathrm{b}\mathcal{C}$ is mapped to $\mathcal{C}(A,B)$.
• On morphisms: Given objects $B,C\in \mathrm{O}\mathrm{b}\mathcal{C}$ and an element $f\in \mathcal{C}(A,B)$, $f$ gets sent to the map $\mathcal{F}f:\mathcal{C}(A,B)\to \mathcal{C}(A,C)$ defined by:

$\mathcal{F}f=g\mapsto f\circ g$.

A set-valued functor is termed representable if it is naturally isomorphic to a covariant Hom functor.

Facts

• The covariant Hom functor is actually a functor. For full proof, refer: Covariant Hom functor is functor
• A morphism between two objects in $\mathcal{C}$ induces a natural transformation between the corresponding covariant Hom functors. For full proof, refer: Morphism induces natural transformation of induced covariant Hom functors