Monadic cointegrals and applications to quasi-Hopf algebras

Understand

For $\mathcal{C}$ a finite tensor category we consider four versions of the central monad, $A_1, \dots, A_4$ on $\mathcal{C}$.

• Two of them are Hopf monads, and for $\mathcal{C}$ pivotal, so are the remaining two.
• In that case all $A_i$ are isomorphic as Hopf monads.
• We define a monadic cointegral for $A_i$ to be an $A_i$-module morphism $\mathbf{1} \to A_i(D)$, where $D$ is the distinguished invertible object of $\mathcal{C}$.