Monoidal monad

In category theory, a monoidal monad formula_1 is a monad formula_2 on a monoidal category formula_3 such that the functor formula_4 is a lax monoidal functor and the natural transformations formula_5 and formula_6 are monoidal natural transformations. In other words, formula_7 is equipped with coherence maps formula_8 and formula_9 satisfying certain properties (again: they are lax monoidal), and the unit formula_10 and multiplication formula_11 are monoidal natural transformations. By monoidality of formula_5, the morphisms formula_13 and formula_14 are necessarily equal.
All of the above can be compressed into the statement that a monoidal monad is a monad in the 2-category formula_15 of monoidal categories, lax monoidal functors, and monoidal natural transformations.
Opmonoidal Monads.
Opmonoidal monads have been studied under various names. Ieke Moerdijk introduced them as "Hopf monads", while in works of Bruguières and Virelizier they are called "bimonads", by analogy to "bialgebra", reserving the term "Hopf monad" for opmonoidal monads with an antipode, in analogy to "Hopf algebras".
An opmonoidal monad is a monad formula_2 "in" the 2-category of formula_17 monoidal categories, oplax monoidal functors and monoidal natural transformations. That means a monad formula_2 "on" a monoidal category formula_3 together with coherence maps formula_20 and formula_21 satisfying three axioms that make an opmonoidal functor, and four more axioms that make the unit formula_5 and the multiplication formula_6 into opmonoidal natural transformations. Alternatively, an opmonoidal monad is a monad on a monoidal category such that the category of Eilenberg-Moore algebras has a monoidal structure for which the forgetful functor is strong monoidal.
An easy example for the monoidal category formula_24 of vector spaces is the monad formula_25, where formula_26 is a bialgebra. The multiplication and unit of formula_26 define the multiplication and unit of the monad, while the comultiplication and counit of formula_26 give rise to the opmonoidal structure. The algebras of this monad are right formula_26-modules, which one may tensor in the same way as their underlying vector spaces.
Examples.
The following monads on the category of sets, with its cartesian monoidal structure, are monoidal monads:
The following monads on the category of sets, with its cartesian monoidal structure, are "not" monoidal monads