Full faithfulness from counit coequalizers ($FGFGB\rightrightarrows FGB\stackrel{{\epsilon }_B}{\to }B$)

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 ${\epsilon }_{C}F\alpha$ equalize $FG{\epsilon }_B$ and ${\epsilon }_{FGB}$; its unique descent is the required morphism $B\to C$. The triangle identities for an adjunction make $G{\epsilon }_B$ a split epimorphism, so descent has underlying arrow exactly $\alpha$. Epimorphic cancellation at ${\epsilon }_B$ proves faithfulness.

For an adjunction $F⊣G$, applying $G$ to its standard counit presentation gives $T^{2}GD\rightrightarrows TGD\to GD$, split by the monad units. Reflection therefore makes every counit presentation a coequalizer and gives full faithfulness by descent. Conversely, monad algebra forgetful functor creates split coequalizers makes an existing G-split quotient a coequalizer in the Eilenberg-Moore category. Fullness and faithfulness of the comparison lift its universal factorization back to the original category.