isomorphisms under functors

Let \(\mathcal{C}, \mathcal{D}\) be categories and \(f \in \mathrm{Hom}_{\mathcal{C}}(X,Y)\) be an Isomorphism and \(F: \mathcal{C} \rightarrow \mathcal{D}\) a functor. Then \(F(f)\) is also an Isomorphism

Note, that given a morphism \(f\) and functor \(\mathcal{F}\), if \(\mathcal{F}(f)\) is not an isomorphism, then \(f\) is also not an isomorphism

By assumption, there exists an Isomorphism \(f^{-1}: Y \rightarrow X\) such that \(f \circ f^{-1} = \mathrm{id}_{Y}\) We know, that \(F(\mathrm{id}_{X}) = \mathrm{id}_{F(x)}\). Therefore, we conclude:

\begin{align*} \mathrm{id}_{F(Y)} = F(f \circ f^{-1}) = F(f) \circ F(f^{-1}) \end{align*}

and \(F(f)\) is an Isomorphism