Zoran Skoda
Beck's theorem

Special case of Beck theorem. Let $Q^{*}⊣Q_{*}$ be an adjoint pair $T$ its associated monad, and $G$ its associated comonad.

If $Q_{*}$ preserves and reflects coequalizers of all parallel pairs in $A$ (for which coequalizers exists) and if any parallel pair mapped by $Q_{*}$ into a pair having a coequalizer in $B$ has a coequalizer in $A$, then the comparison functor $K:B\to A^T$ is an equivalence of categories.

If $Q^{*}$ preserves and reflects equalizers of all parallel pairs in $B$ (for which equalizers exists) and if any parallel pair mapped by $Q^{*}$ into a pair having an equalizer in $A$ has an equalizer in $B$, then the comparison functor $K\prime :A\to G-\mathrm{Comod}$ is an equivalence of categories.

See also monadic functor, monadic adjunction.