Monoidal Monad

In category theory, a monoidal monad is a monad on a monoidal category such that the functor

is a lax monoidal functor with

and

as coherence maps, and the natural transformations

and

are monoidal natural transformations.

By monoidality of, the morphisms and are necessarily equal.

This is equivalent to saying that a monoidal monad is a monad in the 2-category MonCat of monoidal categories, monoidal functors, and monoidal natural transformations.