= Full faithfulness from counit coequalizers
{title2=$FGFGB\rightrightarrows FGB\xrightarrow{\varepsilon_B}B$}
If each standard counit presentation is a <coequalizer>, the <Eilenberg-Moore comparison functor> is <full and faithful>. An <monad algebra morphism> $\alpha:GB\to GC$ makes $\varepsilon_CF\alpha$ equalize $FG\varepsilon_B$ and $\varepsilon_{FGB}$; its unique descent is the required <morphism> $B\to C$. The <triangle identities for an adjunction> make $G\varepsilon_B$ a <split epimorphism>, so descent has underlying arrow exactly $\alpha$. Epimorphic cancellation at $\varepsilon_B$ proves faithfulness.
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