This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (February 2017) ( Learn how and when to remove this message)

This article does not cite any sources. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources:  "Monoidal adjunction" –  news · newspapers · books · scholar · JSTOR (March 2009) ( Learn how and when to remove this message)

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.

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 ${\displaystyle F:{\mathcal {C}}\to {\mathcal {D}}}$ 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.

See also

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

References