Monoidal adjunction

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Suppose that $(\mathcal$ and $(\mathcal$ are two monoidal categories. A monoidal adjunction between two lax monoidal functors

$(F,m):(\mathcal$ and $(G,n):(\mathcal$

is an adjunction $(F, G,η,ε)$ between the underlying functors, such that the natural transformations

$\eta:1_{\mathcal$ and $\varepsilon:F\circ$

are monoidal natural transformations.

[edit] Lifting adjunctions to monoidal adjunctions

Suppose that

$(F,m):(\mathcal$

is a lax monoidal functor such that the underlying functor $F:\mathcal$ has a right adjoint $G:\mathcal$ . This adjuction lifts to a monoidal adjuction $(F, m)$ ⊣ $(G, n)$ if and only if the lax monoidal functor $(F, m)$ is strong.

[edit] See also

Every monoidal adjunction $(F, m)$ ⊣ $(G, n)$ defines a monoidal monad $G\circ$ .