Idea
In category theory, a natural isomorphism is an invertible natural transformation between pairs of functors. The naturality condition makes it a 2-isomorphism in the category Cat.
Definition
Let be categories. Let be functors. A natural isomorphism is a family of isomorphism such that, for all pair of objects such that the following naturality square commutes:
\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
F x \arrow[d, "\alpha_x"'] \arrow[r, "Ff"] & F y \arrow[d, "\alpha_y"] \\
G x \arrow[r, "Gf"'] & G y
\end{tikzcd}
\end{document}