Category of Functors

Let and be categories
Then is the Category of Functors
and Natural Transformations between them.

Theorem

The above definition is correct, i.e. is always a Category

Proof

Composition

Given Functors
and Natural Transformations and
the composition between them
is given by sending to
For any Morphism in
the following is a Commutative Diagram:

which we obtained by combining Naturality Squares of and at .
Thus

and hence is a Natural Transformation.

Identity

Given a Functor
we have the identity Natural Transformation
sending each to
This clearly respects composition.

Associativity

Given Functors
and Natural Transformations

for any object we have:

by Associativity of Morphisms in .
Thus indeed: