Nakayama functor for monads on finite abelian categories

If $\mathcal{M}$ is a finite abelian category and $\mathbf{T}$ is a linear right exact monad on $\mathcal{M}$, then the category $\mathbf{T}-mod$ of $\mathbf{T}$-modules is a finite abelian category. We give an explicit formula of the Nakayama functor of $\mathbf{T}-mod$ under the assumption that the underlying functor of the monad $\mathbf{T}$ has a double left adjoint and a double right adjoint. As applications, we deduce formulas of the Nakayama functor of the center of a finite bimodule category and the dual of a finite tensor category. Some examples from the Hopf algebra theory are also discussed.