= Solution
For an adjunction $F\dashv U:\mathcal D\to\mathcal C$ with induced monad $T=UF$, the <Eilenberg-Moore comparison functor> is
$$
K:\mathcal D\to\mathcal C^T,\qquad
K(D)=(UD,U\varepsilon_D).
$$
The adjunction is <monadic adjunction>[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.
Back to article page
