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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?

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  • $\begingroup$ The $J_{X, Y}$ need to satisfy certain compatibility conditions which are laid out e.g. on Wikipedia: en.wikipedia.org/wiki/Monoidal_functor#Definition $\endgroup$ – Qiaochu Yuan Aug 16 '12 at 17:00
  • $\begingroup$ @QiaochuYuan: I believe the OP assumed these conditions when writing that the $J_{X,Y}$ "give $F$ the structure of a monoidal functor". $\endgroup$ – bradhd Aug 16 '12 at 17:06
  • $\begingroup$ Yes, I assumed the compatibility conditions. Thanks for clarifying this though. $\endgroup$ – MikhailMatrix Aug 16 '12 at 17:12
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    $\begingroup$ Your argument looks good so far, but to be complete you should perhaps verify that these isomorphisms you produced satisfy the aforementioned compatibility conditions themselves. It would be pretty tedious, but I believe it works out. $\endgroup$ – bradhd Aug 16 '12 at 17:19
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I think you want to be a little more careful, as a priori there might be multiple ways of writing a given object as F(X).

This isn't a really problem though because if you have some object A, you know basically how to write it as F(X): just let X = G(A)! So we just have $$G(A \bullet B) \cong G(F(G(A))\bullet F(G(B))) \cong GF(G(A) \otimes G(B)) \cong G(A) \otimes G(B).$$

Again, as pointed out in the comments you need to check that this map really does satisfy the hexagon relations, but that's straightforward.

(Bonus exercise: Use a similar argument to see that the left adjoint of a strong monoidal functor is lax monoidal, while the right adjoint of a strong monoidal functor is oplax monoidal.)

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