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In mathematics, a monoidal adjunction is an adjunction between monoidal categories which respects their monoidal structures.

Suppose that $({\mathcal {C}},\otimes ,I)$ and $({\mathcal {D}},\bullet ,J)$ are two monoidal categories. A monoidal adjunction between two lax monoidal functors

$(F,m):({\mathcal {C}},\otimes ,I)\to ({\mathcal {D}},\bullet ,J)$ and $(G,n):({\mathcal {D}},\bullet ,J)\to ({\mathcal {C}},\otimes ,I)$

is an adjunction $(F,G,\eta ,\varepsilon )$ between the underlying functors, such that the natural transformations

$\eta :1_{\mathcal {C}}\Rightarrow G\circ F$ and $\varepsilon :F\circ G\Rightarrow 1_{\mathcal {D}}$

are monoidal natural transformations.

Lifting adjunctions to monoidal adjunctions

Suppose that

$(F,m):({\mathcal {C}},\otimes ,I)\to ({\mathcal {D}},\bullet ,J)$

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

See also

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