= 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 $q:GB\to Q$, $s:Q\to GB$, $t:GB\to GA$ with
$$
qGf=qGg,\qquad qs=1_Q,\qquad(Gf)t=1_{GB},\qquad(Gg)t=sq.
$$
These 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.
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