left-inverse morphism

Let \(\mathcal{C}\) be a category and \(f \in \mathrm{Hom}_{\mathcal{C}}(A,B)\). A left-inverse morphism \(g\) of \(f\) is a morphism \(g \in \mathrm{Hom}_{\mathcal{C}}(B,A)\) such that

\begin{align*} g \circ f = \mathrm{id}_{A} \end{align*}