Opmonoidal monad ($(T,\eta ,\mu ,T_2,T_0)$)

A monad whose endofunctor is an opmonoidal functor and whose unit and multiplication are opmonoidal natural transformations. Its Eilenberg-Moore category has a tensor of algebras $(X,a)\otimes (Y,b)=(X\otimes Y,(a\otimes b)T_2)$ and unit $(I,T_0)$.