\begin{definition}[Definition:Epi-Mono Factorization]
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
:$f = m \circ e$
Category:Definitions/Category Theory
\end{definition}