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?



Equivalence of definitions tensor functor
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.


