# Monoidal functor and the units

In their book Tensor Categories Etingof, Gelaki, Nikshych and Ostrik define a monoidal functor between monoidal categories $$(C,\otimes,1,\alpha,r,s)$$ and $$(C',\otimes', 1',\alpha',r',s')$$ as the functor $$F\colon C\to C'$$ together with a natural isomorphism $$J_{X,Y}\colon F(X)\otimes'F(Y)\to F(X\otimes Y)$$ such that $$F(1), 1'$$ are isomorphic and the diagram

commutes.

Unlike other sources, they do not require a canonical isomorphism $$F(1)\to 1'$$ for which certain diagrams commute as a part of the data, but claim that it is possible to prove that such an isomorphism exists (provided there exists some isomorphism between them):

However, I find it troubling to prove that such an isomorphism exists.

The diagram they give defines a canonical morphism $$\psi: 1'\otimes' F(1)\to F(1)\otimes' F(1)$$.
Now you can use the following fact: in a monoidal category, if $$A$$ is isomorphic to the unit, then for any morphism $$g: X\otimes A\to Y\otimes A$$ there exists a unique morphism $$f: X\to Y$$ such that $$g=f\otimes Id_A$$. You can apply this with $$A=F(1)$$, $$X=1'$$, $$Y=F(1)$$ and $$g=\psi$$. The point is that you don't need to specify the isomorphism between $$A$$ and the unit, just its existence is enough.