Monomorphism that is Split Epimorphism is Split Monomorphism

Theorem
Let $\mathbf C$ be a metacategory.

Let $f: C \to D$ be a monomorphism and a split epimorphism.

Then $f: C \to D$ is a split epimorphism.

Proof
Let $g: D \to C$ be the right inverse of $f$, i.e:


 * $f \circ g = \operatorname{id}_D$

which is guaranteed to exist by definition of split epimorphism. Then


 * $f \circ g \circ f = \operatorname{id}_D \circ f = f \circ \operatorname{id}_C$

since $f$ is left cancellable, by the definition of monomorphism, we have:


 * $g \circ f = \operatorname{id}_C$

Hence $f$ is a split monomorphism with left inverse $g$.