Covariant Hom functor

Definition

Suppose ${\mathcal {C}}$ is a locally small category and $A\in \operatorname {Ob} {\mathcal {C}}$. 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 $B\in \operatorname {Ob} {\mathcal {C}}$ is mapped to ${\mathcal {C}}(A,B)$.
• On morphisms: Given objects $B,C\in \operatorname {Ob} {\mathcal {C}}$ 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