When an equivalence is a monoidal equivalence?

Question

I want to understand the followings from "Tensor Categories" by Etingof, Gelaki, Nikshych, and Ostrik.

Remark 1.5.3 It is easy to show that if $F: C \to C'$ is an equivalence of monoidal categories, then there exists a monoidal equivalence $F^{-1}: C' \to C$ such that the functors $F \circ F^{-1}$ and $F^{-1} \circ F$ are isomorphic to the identity functor as monoidal functors.

As indicated in the answer to the question, not every equivalence is a monoidal equivalence.

So what I am missing? Does Remark 1.5.3 contain more information than just a equivalence so that I can prove it is a monoidal equivalence? Or I am confused by some definitions?

Thanks.

Answer 1

The meaning of "equivalence" is context-sensitive, just as for "isomorphism". Here, it is clear that "equivalence of monoidal categories" means the same as "monoidal equivalence".

Answer 2

The authors define: "A monoidal functor F is said to be an equivalence of monoidal categories if it is an equivalence of ordinary categories." I.e. a monoidal equivalence F is an equivalence, and F is equipped with a monoidal structure J. Since F is an equivalence, it admits a quasi-inverse G such that FG and GF are isomorphic to the respective identity functors.

What is less obvious (to a beginner like myself) is that G also may equipped with its own monoidal structure.