Monadic cointegrals and applications to quasi-Hopf algebras

For C a finite tensor category we consider four versions of the central monad, A1,…,A4 on C. Two of them are Hopf monads, and for C pivotal, so are the remaining two. In that case all Ai are isomorphic as Hopf monads. We define a monadic cointegral for Ai to be an Ai-module morphism 1→Ai(D), where D is the distinguished invertible object of C.