If we consider a semisimple linear monoidal category $\mathcal{C}$, $\mathcal{D}$ another linear monoidal category, a linear functor $F:\mathcal{C}\rightarrow \mathcal{D}$ and natural isomorphisms $J_{X,Y}: F(X)\otimes F(Y)\rightarrow F(X\otimes Y)$. If I want to show that the functor $(F,J_{X,Y})$ satisfies the monoidal structure axiom is it enough to just check it holds on the simple objects of $\mathcal{C}$ ? I feel like this should be true but I can't seem to show it because I would have to keep track of the isomorphisms from any object $Y$ into its direct sum of simple objects.
Moreover, if I consider $\mathcal{C}$ to be a category which is tensor and linearly generated by just one object (for example the category of graded vector spaces over $\mathbb{Z}/n\mathbb{Z}$) is a fully faithful functor uniquely defined by its value at the generator? (at least up to natural monoidal isomorphism?)