Show if $e : B \rightarrow C$ is the coequalizer of two parallel morphisms $f,g: A \rightarrow B$
and $w: W \rightarrow A$ is epic, then $e$ is the coequalizer of parallel morphisms $fw,gw: W \rightarrow A$
Evidently $efw=egw$. Next let $e'fw=e'gw$ for some $e':B\to C'$. Since $w$ is epic it implies $e'f=e'g$. Since $e$ is a coequalizer, there is unique $k:C\to C'$ such that $ke=e'$ as required.