[proofplan] We must prove that the componentwise inverse family has the correct source and target and satisfies the naturality square for every morphism in $C$. The source and target follow from the definition of inverse morphism. For naturality, we start from the naturality equation for $\eta$ and compose with the inverse components on the appropriate sides. Associativity of composition and the inverse identities reduce the equation to the naturality equation for $\eta^{-1}$. [/proofplan] [step:Define the candidate inverse family with the required component types] For each object $X\in \operatorname{Ob}(C)$, the hypothesis says that \begin{align*} \eta_X:F(X)\to G(X) \end{align*} is an isomorphism in $D$. Hence there exists a morphism \begin{align*} \eta_X^{-1}:G(X)\to F(X) \end{align*} in $D$ satisfying \begin{align*} \eta_X^{-1}\circ \eta_X &= \operatorname{id}_{F(X)},& \eta_X\circ \eta_X^{-1} &= \operatorname{id}_{G(X)}. \end{align*} Thus the family $(\eta_X^{-1})_{X\in \operatorname{Ob}(C)}$ has the component types required for a natural transformation $G\Rightarrow F$. [/step] [step:Derive the naturality equation for the inverse family] Let $f:X\to Y$ be a morphism in $C$. Naturality of $\eta:F\Rightarrow G$ gives the equality in $D$ \begin{align*} G(f)\circ \eta_X=\eta_Y\circ F(f). \end{align*} Compose this equality on the left with $\eta_Y^{-1}:G(Y)\to F(Y)$ and on the right with $\eta_X^{-1}:G(X)\to F(X)$. Associativity of composition in $D$ gives \begin{align*} \eta_Y^{-1}\circ G(f)\circ \eta_X\circ \eta_X^{-1} = \eta_Y^{-1}\circ \eta_Y\circ F(f)\circ \eta_X^{-1}. \end{align*} Using the inverse identities \begin{align*} \eta_X\circ \eta_X^{-1} &= \operatorname{id}_{G(X)},& \eta_Y^{-1}\circ \eta_Y &= \operatorname{id}_{F(Y)}, \end{align*} and the identity laws in $D$, this becomes \begin{align*} \eta_Y^{-1}\circ G(f)=F(f)\circ \eta_X^{-1}. \end{align*} Equivalently, \begin{align*} F(f)\circ \eta_X^{-1}=\eta_Y^{-1}\circ G(f). \end{align*} This is precisely the naturality equation for the family $(\eta_X^{-1})_{X\in \operatorname{Ob}(C)}$ as a transformation $G\Rightarrow F$. [/step] [step:Conclude that the componentwise inverse is a natural transformation] Since the preceding naturality equation holds for every morphism $f:X\to Y$ in $C$, the family $(\eta_X^{-1})_{X\in \operatorname{Ob}(C)}$ is natural in $X$. Therefore it defines a natural transformation \begin{align*} \eta^{-1}:G\Rightarrow F, \end{align*} whose component at each object $X$ is $\eta_X^{-1}:G(X)\to F(X)$. [/step]