Suppose that ( C , ⊗ , I ) and ( D , ∙ , J ) are two monoidal categories. A monoidal adjunction between two lax monoidal functors
( F , m ) : ( C , ⊗ , I ) → ( D , ∙ , J ) and
( G , n ) : ( D , ∙ , J ) → ( C , ⊗ , I ) is an adjunction ( F , G , η , ε ) between the underlying functors, such that the natural transformations
η : 1 C ⇒ G ∘ F and
ε : F ∘ G ⇒ 1 D are monoidal natural transformations.
Suppose that
( F , m ) : ( C , ⊗ , I ) → ( D , ∙ , J ) is a lax monoidal functor such that the underlying functor F : C → D has a right adjoint G : D → C . This adjuction lifts to a monoidal adjuction ( F , m ) ⊣ ( G , n ) if and only if the lax monoidal functor ( F , m ) is strong.
