Monoidal adjunction

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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[edit]

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[edit]

  • Every monoidal adjunction defines a monoidal monad .