monic split epi as isomorphism

Let \(\mathcal{C}\) be a category, \(f\) a monomorphism and split epi. Then \(f\) is an isomorphism

By assumption, there exists a right-inverse \(f^{-1} \in \mathrm{Hom}_{\mathcal{C}}(B,A)\) such that $f ˆ f^-1 = \mathrm{id}_B. Furthermore

\begin{align*} \mathrm{id}_{B} \circ f =& f \circ f^{-1} \circ f \\ (f \circ f^{-1}) \circ f =& \mathrm{id}_{A} \circ f \end{align*}

and since \(f\) is monic, it follows that

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

and thus \(f\) is an iso