 Reflexive coequalizer

A reflexive coequalizer is a coequalizer of a reflexive pair, meaning a parallel pair $r,s:X\rightrightarrows Y$ with a common section $t:Y\to X$ satisfying $rt=st=1_Y$. A functor preserves reflexive coequalizers when it takes every such coequalizer to a coequalizer of the image pair.