# 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?