Monoidal adjunction

Suppose that $(\mathcal C,\otimes,I)$ 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 $(F,G,\eta ,\varepsilon )$ 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.

Lifting adjunctions to monoidal adjunctions

Suppose that

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

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