Coequalizer

The coequalizer of parallel arrows $f,g:A\rightrightarrows B$ is a universal arrow $q:B\to Q$ satisfying $qf=qg$. Every other arrow equalizing $f$ and $g$ factors uniquely through $q$.

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.

For $G:\mathcal{D}\to \mathcal{C}$, a parallel pair in $\mathcal{D}$ is G-split if its image admits a split coequalizer in $\mathcal{C}$. Reflection of such coequalizers means that an existing equalizing arrow in $\mathcal{D}$ is a coequalizer whenever its image is the corresponding split coequalizer. This asserts a property of existing arrows; it does not assert that a lift exists for every base quotient.