ct.category theory – Equivalence of definitions tensor functor

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 $J_{X,Y}: F(X)otimes’ F(Y) to F(Xotimes Y)$ 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 $theta: F’Fto operatorname{id}_{mathcal{C}}$.

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?