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
\begin{align*} \alpha ^{-1}\circ \alpha =\text{id}_{F},\\ \alpha \circ \alpha ^{-1}=\text{id}_{G}. \end{align*}
The implication Item (a)$\implies $Item (b) is clear, whereas the implication Item (b)$\implies $Item (a) follows from Item 2.
Item 2: Componentwise Inverses of Natural Transformations Assemble Into Natural Transformations
The naturality condition for $\alpha ^{-1}$ corresponds to the commutativity of the diagram
for each $A,B\in \text{Obj}\webleft (\mathcal{C}\webright )$ and each $f\in \textup{Hom}_{\mathcal{C}}\webleft (A,B\webright )$. Considering the diagram
where the boundary diagram as well as Subdiagram (2) commute, we have
\begin{align*} G\webleft (f\webright ) & = G\webleft (f\webright )\circ \text{id}_{G\webleft (A\webright )}\\ & = G\webleft (f\webright )\circ \alpha _{A}\circ \alpha ^{-1}_{A}\\ & = \alpha _{B}\circ F\webleft (f\webright )\circ \alpha ^{-1}_{A}. \end{align*}
Postcomposing both sides with $\alpha ^{-1}_{B}$, we get
\begin{align*} \alpha ^{-1}_{B}\circ G\webleft (f\webright ) & = \alpha ^{-1}_{B}\circ \alpha _{B}\circ F\webleft (f\webright )\circ \alpha ^{-1}_{A}\\ & = \text{id}_{F\webleft (B\webright )}\circ F\webleft (f\webright )\circ \alpha ^{-1}_{A}\\ & = F\webleft (f\webright )\circ \alpha ^{-1}_{A}, \end{align*}
which is the naturality condition we wanted to show. Thus $\alpha ^{-1}$ is a natural transformation.
