Suppose that and are two monoidal categories

Monoidal category

In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...

. A monoidal adjunction between two lax monoidal functors and
is an adjunction  between the underlying functors, such that the natural transformation

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

s and
are monoidal natural transformation

Monoidal natural transformation

Suppose that and are two monoidal categories and:\to and :\toare two lax monoidal functors between those categories....

s.

Lifting adjunctions to monoidal adjunctions

Suppose that
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.

See also

• Every monoidal adjunction ⊣ defines a monoidal monad .