Monoidal adjunction

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.

Monoidal adjunction

Lifting adjunctions to monoidal adjunctions

See also