 Split coequalizer

For parallel arrows $f,g:B\rightrightarrows C$, a split coequalizer consists of $q:C\to Q$ and arrows $s:Q\to C$, $t:C\to B$ with $qf=qg$, $qs=1_Q$, $ft=1_C$, and $gt=sq$. These equations prove the coequalizer property: if $uf=ug$, then $u=uft=ugt=usq$, so $us$ is the unique factor through $q$. Every functor preserves this split coequalizer because it preserves these equations.