split epi as epimorphism

Let \(\mathcal{C}\) be a category and \(f\) a split epi Then \(f\) is also an epimorphism

By assumption there exists a right-inverse morphism \(f^{-1}\) Suppose \(g \circ f = h \circ f\), then by assumption we get

\begin{align*} (g \circ f) \circ f^{-1} =& (h \circ f) \circ f^{-1} \\ g \circ \mathrm{id} =& h \circ \mathrm{id} \\ g =& h \end{align*}