isomorphism

Let \(\mathcal{C}\) be a category, \(A,B \in \mathrm{Ob}(\mathcal{C})\) and \(f \in \mathrm{Hom}_{\mathcal{C}}(A,B)\).

Then \(f\) is an isomorphism, if it is both split mono and split epi, i.e. there exist a left-inverse \(f_l^{-1}\) and a right inverse \(f_{r}^{-1}\).

note, that \(f_{r}^{-1} = f_l^{-1}\) (uniqueness of a two-sided inverse morphism)