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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.

Assume the equivalent conditions and that $T$ preserves finite coproducts. The free functor already reflects isomorphisms. By the dual Crude monadicity theorem, it remains to preserve the relevant coreflexive equalizers.

A coreflexive equalizer diagramwith $A\ne \varnothing$ can be equipped with the extra sections making it a split equalizer: choose a point of $A$ and use the common retraction to define the missing splitting maps on the complementary fibres. Every functor preserves split equalizers. If $A=\varnothing$, preservation follows from preservation of the initial object, which follows from preservation of finite coproducts. Consequently $F$ preserves all required coreflexive equalizers, and the adjunction is comonadic.