= Surjection-embedding factorization of a geometric morphism
{title2=$\mathcal E\xrightarrow{p}\mathcal D\xrightarrow{i}\mathcal F$}
For $f:\mathcal E\to\mathcal F$, the comonad $G=f^*f_*$ is Cartesian, and its coalgebra topos $\mathcal D$ gives a factorization $f=i\circ p$. Here $p^*$ is the faithful forgetful functor and $i^*$ is the comparison functor. Its right adjoint sends $(A,a)$ to $\operatorname{Eq}(f_*a,\eta_{f_*A})$; applying $f^*$ identifies the comparison counit with the equalizer $a:A\to GA$ of $Ga,\delta_A$, proving that $i_*$ is full and faithful.
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