Consider the following: two monoidal categories $({\cal C},\otimes)$, and $({\cal D},\odot)$, and a functor $F:{\cal C} \to {\cal D}$, that gives an equivalence (of ordinary categories) between ${\cal C}$ and ${\cal D}$. Moreover, let us assume that we also have $ F(X \otimes Y) = F(X) \odot F(Y), ~~~~~ \text{ for } X,Y \in {\cal C}. $ Does it automatically follow from this that $F$ is a (strict) equivalence of mondoidal categories? That is to say, do the necessary coherence conditions hold automatically?
To me this seems obviously true. However, there have times when what were "obvious truths" to me in general category theory, turned out to be anything but. So, I'd like a voice of confirmation. Thanks!