Let $F,G:\mathcal{C}\to \mathcal{D}$ be functors and let $\alpha:F\Rightarrow G$ be a natural transformation between them. Suppose that, for every object $C\in\mathcal{C}$, the morphism $\alpha_C:FC\to GC$ is an isomorphism, and let $\beta_C:GC\to FC$ be its inverse. I want to prove that $\beta=(\beta_C:GC\to FC)_{C\in\mathcal{C}}$ is a natural transformation. To do that, for every morphism $h:C\to C'$ in $\mathcal{C}$, I need to show that $Fh\circ \beta_C=\beta_{C'}\circ Gh$. What I know is that $Gh\circ\alpha_C=\alpha_{C'}\circ Fh$.
Can you give some help please?
Best Answer
Just use the naturality relation and the fact that $\alpha^{-1}=\beta.$ That is,
$\alpha_{C'}\circ Fh\circ \beta_C=Gh\circ\alpha_C\circ\beta_C=Gh\circ1_C=Gh\Rightarrow Fh\circ \beta_C=\beta_{C'}\circ Gh$