Monoidal adjunction

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A monoidal adjunction is an adjunction in mathematics between monoidal categories which respects the monoidal structure.^[1][2][3]

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

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

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

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

are monoidal natural transformations.

Suppose that

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

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

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