A natural transformation $\alpha \colon F\Longrightarrow G$ is a natural isomorphism if there exists a natural transformation $\alpha ^{-1}\colon G\Longrightarrow F$ such that
Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $F,G\colon \mathcal{C}\rightrightarrows \mathcal{D}$ be functors.
A natural transformation $\alpha \colon F\Longrightarrow G$ is a natural isomorphism if there exists a natural transformation $\alpha ^{-1}\colon G\Longrightarrow F$ such that
Let $\alpha \colon F\Longrightarrow G$ be a natural transformation.
Componentwise Inverses of Natural Transformations Assemble Into Natural Transformations. Let $\alpha ^{-1}\colon G\Longrightarrow F$ be a transformation such that, for each $A\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$, we have
Then $\alpha ^{-1}$ is a natural transformation.
Postcomposing both sides with $\alpha ^{-1}_{B}$, we get
which is the naturality condition we wanted to show. Thus $\alpha ^{-1}$ is a natural transformation.