Let $\mathcal{C}$ and $\mathcal{D}$ be two semisimple abelian categories. Is it true that $\mathcal{C}$ and $\mathcal{D}$ are equivalent IFF they the cardinality of their classes of simple objects is the same.
This seems like it should be obviously true, but I am afraid I miss something obvious.
No; you also need the isomorphism classes of the endomorphism rings to match.