Suppose that and are two monoidal categories. A monoidal adjunction between two lax monoidal functors
is an adjunction between the underlying functors, such that the natural transformations
are monoidal natural transformations.
Lifting adjunctions to monoidal adjunctions
is a lax monoidal functor such that the underlying functor has a right adjoint . This adjuction lifts to a monoidal adjuction ⊣ if and only if the lax monoidal functor is strong.
- Every monoidal adjunction ⊣ defines a monoidal monad .