Monomorphism that is Split Epimorphism is Split Monomorphism
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Theorem
Let $\mathbf C$ be a metacategory.
Let $f: C \to D$ be a morphism in $\mathbf C$ such that $f$ is a monomorphism and a split epimorphism.
Then $f: C \to D$ is a split monomorphism.
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Proof
Let $g: D \to C$ be the right inverse of $f$:
- $f \circ g = \operatorname{id}_D$
which is guaranteed to exist by definition of split epimorphism.
Therefore:
- $f \circ g \circ f = \operatorname{id}_D \circ f = f \circ \operatorname{id}_C$
by the property of the identity morphism.
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 $g$.
$\blacksquare$