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.