Editing Functor category (section)

== Definition ==
Suppose <math>C</math> is a [[small category]] (i.e. the objects and morphisms form a set rather than a [[proper class]]) and <math>D</math> is an arbitrary category. 
The category of functors from <math>C</math> to <math>D</math>, written as Fun(<math>C</math>, <math>D</math>), Funct(<math>C</math>,<math>D</math>), <math>[C,D]</math>, or <math>D ^C</math>, has as objects the covariant functors from <math>C</math> to <math>D</math>, 
and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: 
if <math>\mu (X) : F(X) \to G(X)</math> is a natural transformation from the functor <math>F : C \to D</math> to the functor <math>G : C \to D</math>, and 
<math>\eta(X) : G(X) \to H(X)</math> is a natural transformation from the functor <math>G</math> to the functor <math>H</math>, then the composition <math>\eta(X)\mu(X) : F(X) \to H(X)</math> defines a natural transformation 
from <math>F</math> to <math>H</math>. With this composition of natural transformations (known as vertical composition, see [[natural transformation]]), 
<math>D^C</math> satisfies the axioms of a category.

In a completely analogous way, one can also consider the category of all ''contravariant'' functors from <math>C</math> to <math>D</math>; we write this as Funct(<math>C^\text{op},D</math>).

If <math>C</math> and <math>D</math> are both [[preadditive category|preadditive categories]] (i.e. their morphism sets are [[abelian group]]s and the composition of morphisms is [[bilinear operator|bilinear]]), 
then we can consider the category of all [[additive functor]]s from <math>C</math> to <math>D</math>, denoted by Add(<math>C</math>,<math>D</math>).