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?