Functor category

In category theory, a branch of mathematics, a functor category ${\displaystyle D^{C}}$ is a category where the objects are the functors ${\displaystyle F:C\to D}$ and the morphisms are natural transformations ${\displaystyle \eta :F\to G}$ between the functors (here, ${\displaystyle G:C\to D}$ is another object in the category). Functor categories are of interest for two main reasons:

• many commonly occurring categories are (disguised) functor categories, so any statement proved for general functor categories is widely applicable;
• every category embeds in a functor category (via the Yoneda embedding); the functor category often has nicer properties than the original category, allowing certain operations that were not available in the original setting.