nLab
functor category

Definition

Given categories $C$ and $D$, the functor category – written $D^C$ or $[C,D]$ – is the category whosse

• objects are functors $F:C\to D$

• morphisms are natural transformations between these functors.

Functor categories serve as the hom-categories in the strict 2-category Cat.

In the context of enriched category theory the functor category is generalized to the enriched functor category.

In the absence of the axiom of choice (including many internal situations), the appropriate notion to use is the anafunctor category.