Equivalence of definitions tensor functor

Question

Let $(\mathcal{C}, \otimes , I)$ and $(\mathcal{C}, \otimes', I')$ be tensor categories. A tensor functor $F: (\mathcal{C}, \otimes , I)\to (\mathcal{C}', \otimes' , I')$ consists of a functor $F: \mathcal{C}\to \mathcal{C'}$ together with natural isomorphisms $J_{X,Y}: F(X)\otimes' F(Y) \to F(X\otimes Y)$ and an isomorphism $\varphi: F(I)\to I'$ such that three compatibility diagrams with respect to the associators commute.

Next, consider the following two definitions:

(1) A tensor functor $F$ is called an equivalence of tensor categories if it is an equivalence of ordinary categories.

(2) A tensor functor $F: (\mathcal{C}, \otimes , I)\to (\mathcal{C}', \otimes' , I')$ is called an equivalence of tensor categories if there exists a tensor functor $F': (\mathcal{C}', \otimes' , I')\to (\mathcal{C}, \otimes , I)$ together with natural tensor isomorphisms $\eta: \operatorname{id}_{\mathcal{C'}}\to FF'$ and $\theta: F'F\to \operatorname{id}_{\mathcal{C}}$.

Definition (1) is the definition in Etingof's book "Tensor categories" and definition (2) is in Kassel's book "Quantum groups". Clearly definition (2) implies definition (1). Is the converse true?

Answer 1

Yes. Given (1) with quasi-inverse $F':\mathcal{C}'\to\mathcal{C}$, there is a unique way to make $F'$ a tensor functor such that $\eta$ and $\theta$ are tensor isomorphisms. Specifically, the tensor structure is

$$ F' X \otimes F' Y \cong F' F(F' X \otimes F' Y) \cong F'(F F' X \otimes' F F' Y) \cong F'(X\otimes' Y) $$

and similarly for the unit.

More generally, for any 2-monad $T$ on a 2-category $\mathcal{K}$, a pseudomorphism $f:A\to B$ of $T$-algebras is an internal equivalence in the 2-category of $T$-algebras if and only if its underlying morphism in $\mathcal{K}$ is an equivalence. This is a categorification of the fact that any monadic functor is conservative.