Functor Category - Definition

Definition

Suppose C is a small category (i.e. the objects and morphisms form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Fun(C, D), Funct(C,D) or DC, has as objects the covariant functors from C to D, and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if μ(X) : F(X) → G(X) is a natural transformation from the functor F : C → D to the functor G : C → D, and η(X) : G(X) → H(X) is a natural transformation from the functor G to the functor H, then the collection η(X)μ(X) : F(X) → H(X) defines a natural transformation from F to H. With this composition of natural transformations (known as vertical composition, see natural transformation), DC satisfies the axioms of a category.

In a completely analogous way, one can also consider the category of all contravariant functors from C to D; we write this as Funct(Cop,D).

If C and D are both preadditive categories (i.e. their morphism sets are abelian groups and the composition of morphisms is bilinear), then we can consider the category of all additive functors from C to D, denoted by Add(C,D).