Implication Details

Claim: If a morphism is a strong monomorphism, then it is an extremal monomorphism.

Proof: Assume that $m : A \to B$ is a strong monomorphism that factors as $m = g \circ e$, where $e : A \to C$ is an epimorphism and $g : C \to B$ is any morphism. Then the commutative diagram $$\begin{CD} A @>e>> C \\ @V{\id_A}VV @VV{g}V \\ A @>>m> B \end{CD}$$ can be filled with a morphism $h : C \to A$. In particular, $h \circ e = \id_A$. Thus, $e$ is an epimorphism and a split monomorphism, hence an isomorphism.

This implication has a dual.