Proposition 9.9.7.1.2 (018K): Properties of Natural Isomorphisms

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Let $\alpha \colon F\Longrightarrow G$ be a natural transformation.

Characterisations. The following conditions are equivalent:
1. The natural transformation $\alpha $ is a natural isomorphism.
2. For each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, the morphism $\alpha _{A}\colon F_{A}\to G_{A}$ is an isomorphism.
Componentwise Inverses of Natural Transformations Assemble Into Natural Transformations. Let $\alpha ^{-1}\colon G\Longrightarrow F$ be a transformation such that, for each $A\in \text{Obj}\webleft (\mathcal{C}\webright )$, we have
\begin{align*} \alpha ^{-1}_{A}\circ \alpha _{A} & = \text{id}_{F\webleft (A\webright )},\\ \alpha _{A}\circ \alpha ^{-1}_{A} & = \text{id}_{G\webleft (A\webright )}. \end{align*}

Then $\alpha ^{-1}$ is a natural transformation.

Item 1: Characterisations

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.