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:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
FA \arrow[r,"Ff"] \arrow[d,"\alpha_{A}"]
 & FB \arrow[d,"\alpha_{B}"] \\
GA \arrow[r,"Gf"] \arrow[d,"\beta_{A}"]
 & GB \arrow[d, "\beta_{B}"] \\
 HA \arrow[r, "Hf"]
 & HB
\end{tikzcd}
\end{document}

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: