Commutative graded monads

Abstract: It is well-known that the category of Kleisli algebras for a monoidal monad carries a canonical monoidal structure. We define the notion of a commutative graded monad and present a strictly two-categorical proof that Kleisli algebras for such monads equally carry a canonical monoidal structure which reduce to the monoidal monad case when the commutative graded monad in question is graded over the trivial monoidal category.