 Solution

A functor-split coequalizer pair for $G$ is a parallel pair $f,g:A\rightrightarrows B$ in $\mathcal{D}$ whose image has a split coequalizer in $\mathcal{C}$. Explicitly there are $q:GB\to Q$, $s:Q\to GB$, $t:GB\to GA$ withThese equations give the coequalizer property: if $hGf=hGg$, then $h=h(Gf)t=h(Gg)t=hsq$, and $q$ is split epic so the factor through it is unique. Every functor preserves such a split diagram, since it preserves these equations.

Reflection means that, for such a pair, any existing arrow $e:B\to D$ whose image is a coequalizer is itself a coequalizer in $\mathcal{D}$. A coequalizer of the image pair is isomorphic to the specified split one and inherits its splitting. This is reflection of an existing quotient, not an assertion that every base quotient has a lift.