Editing Natural transformation (section)

=== Vertical composition ===

If <math>\eta : F \Rightarrow G</math> and <math>\epsilon: G \Rightarrow H</math> are natural transformations between functors <math>F, G, H: C \to D</math>, then we can compose them to get a natural transformation <math>\epsilon \circ \eta: F \Rightarrow H</math>. 
This is done component-wise:
:<math>(\epsilon \circ \eta)_X = \epsilon_X \circ \eta_X</math>. 

[[Image:Vertical composition of natural transformations.svg|center|300px]]

This vertical composition of natural transformations is [[associative]] and has an identity, and allows one to consider the collection of all functors <math>C \to D</math> itself as a category (see below under [[#Functor categories|Functor categories]]).
The identity natural transformation <math>\mathrm{id}_F</math> on functor <math>F</math> has components <math>(\mathrm{id}_F)_X = \mathrm{id}_{F(X)}</math>.<ref>{{cite web | url=https://ncatlab.org/nlab/show/identity+natural+transformation | title=Identity natural transformation in nLab }}</ref>
:For <math>\eta : F \Rightarrow G</math>, <math>\mathrm{id}_G \circ \eta = \eta = \eta \circ \mathrm{id}_F</math>.