Natural Isomorphism

Let and be Category
Let be Functors
Let be a Natural Transformation
and suppose is an Isomorphism in the Category of Functors .
Then is called a natural isomorphism.

Lemma

Let be Functors
Let be a Natural Transformation between them.
Then is an Isomorphism in Category of Functors
if and only if
each is an Isomorphism in

Proof

Obvious since composition in is pointwise.

Suppose each has an Inverse
We need to verify naturality of .
Given in , consider the Naturality Square of at :

We have

Thus is a Natural Transformation and by definition:

So is an Isomorphism to .

Special Case

When we get that the following commutes:

i.e. we can write