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 :

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

We have

Thus is a Natural Transformation and by definition:

So is an Isomorphism to .

Special Case

When we get that the following commutes:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A \arrow[r,"f"] \arrow[d, shift left, "\alpha_{A}"]
 & B \arrow[d,shift left, "\alpha_{B}"] \\
GA \arrow[r,"Gf"] \arrow[u, shift left, "\alpha_{A}^{-1}"]
 & GB \arrow[u, shift left, "\alpha_{B}^{-1}"]
\end{tikzcd}
\end{document}

i.e. we can write