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 adjuction lifts to a monoidal adjuction (F,m)⊣(G,n) if and only if the lax monoidal functor (F,m) is strong.

See also