Let ${\cal C,D}$ be two categories, and let $ F:{\cal C} \to {\cal D}, ~~~~~~~~~~~~~~~~~ G:{\cal D} \to {\cal C}, $ be an equivalence of categories. Let us now further assume that ${\cal C}$ can be endowed with a monoidal structure $\otimes$, and that ${\cal D}$ can also be endowed with a monoidal structure $\bullet$. Finally, let us assume, for each $X,Y \in C$, that we have isomorphisms $ J_{X,Y}: F(X \otimes Y) \to F(X) \bullet F(Y), $ that give $F$ the structure of a monoidal functor.
Now it seems to me that this automatically gives $G$ the structure of a monoidal functor: Any two objects in ${\cal D}$ will be isomorphic to $F(X)$, and $F(Y)$, for some $X,Y \in {\cal C}$, and $ G(F(X) \bullet F(Y)) \simeq G(F(X \otimes Y)) \simeq X \otimes Y \simeq G(F(X)) \otimes G(F(Y)), $ giving $G$ the structure of a monoidal functor. Thus, I would conclude that given a equivalence of monoidal categories as simple categories, such that one of the functors in the equivalence is a monoidal functor, then we always get an equivalence of monoidal categories.
Am I correct here, or have I missed some categorical subtlety?