Functor category

Summary

The functor category $[\mathcal{C}, \mathcal{D}]$, also written $\mathcal{D}^{\mathcal{C}}$, has the functors from $\mathcal{C}$ to $\mathcal{D}$ as objects and the natural transformations between them as morphisms. Many familiar categories are functor categories: sets with a group action, directed graphs, and presheaves on a topological space. Its isomorphisms are the natural isomorphisms. Functor categories into the category of sets are the setting of the Yoneda lemma.