Let $\mathcal C,\mathcal D$ be monoidal categories. Recall that a functor $F : \mathcal C \to \mathcal D$ is *lax monoidal* if it is equipped with maps $1\_{\mathcal D} \to F(1\_{\mathcal C})$ and $F(X) \otimes\_{\mathcal D} F(Y) \to F(X \otimes\_{\mathcal C} Y)$, the latter natural in $X,Y\in \mathcal C$, compatible with associators and unitors. It is *oplax monoidal* if it is instead equipped with maps in the other direction, and *strong monoidal* if it is equipped with isomorphisms. For each choice of lax/oplax/strong, there is a bicategory of monoidal categories, lax/oplax/strong monoidal functors, and monoidal natural transformations.


Suppose that $F : \mathcal C \to \mathcal D$ is one of lax/oplax/strong monoidal, and also admits a left adjoint $F^L$ in the bicategory of all functors. Under what circumstances is $F^L$ naturally lax/oplax/strong monoidal? Under what circumstances does this adjunction come from an adjunction in the bicategory of monoidal categories and lax/oplax/strong monoidal functors?



When is the adjoint to a monoidal functor monoidal?
If $L$ and $R$ are a left and right adjoint, then [doctrinal adjunction](http://ncatlab.org/nlab/show/doctrinal+adjunction) asserts that $L$ is oplax monoidal iff $R$ is lax monoidal. (I'm being a bit imprecise here, treating monoidality as if it were a property instead of a structure, but hopefully the meaning is clear.) 


