Definition:Epi-Mono Factorization

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Let $\mathbf C$ be a category.

Let $A$ and $B$ be objects in $\mathbf C$.

Let $f : A \to B$ be a morphism in $\mathbf C$.

An epi-mono-factorization of $f$ is a pair $\struct {e, m}$ such that:

$e : A \to Z$ an epimorphism in $\mathbf C$
$m : Z \to B$ a monomorphism in $\mathbf C$

such that

$f = m \circ e$