Implication details

Proof: Consider a commutative diagram $$\begin{CD} C @>e>> D \\ @VVV @VVV \\ A @>>m> B \end{CD}$$ where $e$ is an epimorphism and $m$ is a strict monomorphism. We need to show that $D \to B$ factors through $m$. It suffices to show that it equalizes all pairs $B \rightrightarrows T$ that are equalized by $m$. Since $e$ is an epimorphism, it suffices to check this for the composite $C \to D \to B$. This is equal to $C \to A \to B$, which factors through $m$ and hence equalizes the pair.