Editing Natural transformation (section)

==Functor categories==

{{Main|Functor category}}
If <math>C</math> is any category and <math>I</math> is a [[small category]], we can form the [[functor category]] <math>C^I</math> having as objects all functors from <math>I</math> to <math>C</math> and as morphisms the natural transformations between those functors. This forms a category since for any functor <math>F</math> there is an identity natural transformation <math>1_F: F \to F</math> (which assigns to every object <math>X</math> the identity morphism on <math>F(X)</math>) and the composition of two natural transformations (the "vertical composition" above) is again a natural transformation.

The [[isomorphism]]s in <math>C^I</math> are precisely the natural isomorphisms. That is, a natural transformation <math>\eta: F \to G</math> is a natural isomorphism if and only if there exists a natural transformation <math>\epsilon: G \to F</math> such that <math>\eta\epsilon = 1_G</math> and <math>\epsilon\eta = 1_F</math>.

The functor category <math>C^I</math> is especially useful if <math>I</math> arises from a [[directed graph]]. For instance, if <math>I</math> is the category of the directed graph {{nobreak|1=• → •}}, then <math>C^I</math> has as objects the morphisms of <math>C</math>, and a morphism between <math>\phi: U \to V</math> and <math>\psi: X \to Y</math> in <math>C^I</math> is a pair of morphisms <math>f: U \to X</math> and <math>g: V \to Y</math> in <math>C</math> such that the "square commutes", i.e. <math>\psi \circ f = g \circ \phi</math>.

More generally, one can build the [[2-category]] <math>\textbf{Cat}</math> whose
* 0-cells (objects) are the small categories,
* 1-cells (arrows) between two objects <math>C</math> and <math>D</math> are the functors from <math>C</math> to <math>D</math>,
* 2-cells between two 1-cells (functors) <math>F:C\to D</math> and <math>G:C\to D</math> are the natural transformations from <math>F</math> to <math>G</math>.
The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category <math>C^I</math> is then simply a hom-category in this category (smallness issues aside).

=== More examples ===
Every [[Limit (category theory)|limit]] and colimit provides an example for a simple natural transformation, as a [[cone (category theory)|cone]] amounts to a natural transformation with the [[diagonal functor]] as domain. Indeed, if limits and colimits are defined directly in terms of their [[universal property]], they are universal morphisms in a functor category.