Monoidal adjunction

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