Natural Transformation

Suppose we are given functors
A natural transformation is an operation
assigning each a Morphism in
such that for each in :

This is equivalent to a Commutative Diagram:

This is called a Naturality Square for at .

Natural transformations are Morphisms in the Category of Functors

Natural Isomorphism
Equivalence

Example

Let be a category with only identity Morphisms.
A Functor is just a sequence in indexed by .
Given , a natural transformation between them
is any assignment for .
If there are no Morphisms for some ,
then there is no natural transformations.

Example

Given Group Actions of a Group on and ,
a natural transformation between them is a -Equivariant
A group action Functor is a functor
sending the only element of to the set that its acting on,
and sending each Morphism to a permutation of .
Suppose and are such functors,
representing Group Actions of on sets and respectively.
A natural transformation is then just a map
such that for any , we have a Commutative Diagram:

i.e. for any :

where represents the Group Action in respective sets.