Surjection-embedding factorization of a geometric morphism ($\mathcal{E}\stackrel{p}{\to }\mathcal{D}\stackrel{i}{\to }\mathcal{F}$)

For $f:\mathcal{E}\to \mathcal{F}$, the comonad $G=f^{\ast }{f}_{\ast }$ is Cartesian, and its coalgebra topos $\mathcal{D}$ gives a factorization $f=i\circ p$. Here $p^{\ast }$ is the faithful forgetful functor and $i^{\ast }$ is the comparison functor. Its right adjoint sends $(A,a)$ to $\mathrm{Eq}(f_{\ast }a,{\eta }_{f_{\ast }A})$; applying $f^{\ast }$ identifies the comparison counit with the equalizer $a:A\to GA$ of $Ga,{\delta }_A$, proving that $i_{\ast }$ is full and faithful.