Comonadic adjunction

An adjunction is comonadic when the comparison from its left-hand category to coalgebras for the induced comonad is an equivalence. The dual Beck theorem tests this using reflected isomorphisms and suitable equalizers.

The relevant criterion is the Beck comonadicity theorem, dual to Beck's monadicity theorem.

The Beck comonadicity theorem, dual to the Beck monadicity theorem, says that a left adjoint is comonadic when it reflects isomorphisms and preserves the required equalizers of pairs whose images admit split equalizers. A stronger convenient hypothesis is that it preserves all equalizers.