Monoidal adjunction


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

Suppose that
is a lax monoidal functor such that the underlying functor has a right adjoint. This adjunction lifts to a monoidal adjunction ⊣ if and only if the lax monoidal functor is strong.