Suppose that and are two monoidal categories. A monoidal adjunction between two lax monoidal functors

and

is an adjunction between the underlying functors, such that the natural transformations

and

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.

See also

  • Every monoidal adjunction defines a monoidal monad .
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