Monoidal adjunction

Suppose that formula_1 and formula_2 are two monoidal categories. A monoidal adjunction between two lax monoidal functors
is an adjunction formula_5 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 formula_9 has a right adjoint formula_10. This adjunction lifts to a monoidal adjunction formula_11⊣formula_12 if and only if the lax monoidal functor formula_11 is strong.