normal epimorphism

Let \(\mathcal{C}\) be a category and \(f \in \mathrm{Hom}_{\mathcal{C}}(B,C)\) a epimorphism for objects \(B,C \in \mathrm{Ob}(\mathcal{C})\). Then \(f\) is said to be normal, if there exists an object \(A \in \mathrm{Ob}(\mathcal{C})\) and a morphism \(g \in \mathrm{Hom}_{\mathcal{C}}(B,C)\), such that \(C\) is isomorphic to the cokernel \(\mathrm{ker}(g)\).