 Power objects turn coreflexive equalizers into coequalizers

For a coreflexive pair $f,g:B\rightrightarrows A$ with equalizer $e:E\hookrightarrow B$, direct image along the mono $f$ satisfies $Pf{\exists }_f=1$ and $Pg{\exists }_f={\exists }_{e}Pe$. Thus any map $h:PB\to Z$ equalizing $Pf,Pg$ obeys $h=h{\exists }_{e}Pe$. Since $Pe{\exists }_e=1$, the map $Pe$ is their coequalizer. These identities hold for parameterized subobjects in any elementary topos.