Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 119 / 4 / a / Solution

For an adjunction $F⊣U:\mathcal{D}\to \mathcal{C}$ with induced monad $T=UF$, the Eilenberg-Moore comparison functor isThe adjunction is monadic when $K$ is an equivalence.

The Crude monadicity theorem states that a right adjoint is monadic if it reflects isomorphisms, its source has coequalizers of reflexive pairs, and it preserves those coequalizers. The dual statement is the crude comonadicity theorem.