Proof.
Since the (surjective on objects and dominant on 1-morphisms, faithful) factorization system is compatible with the monoidal structure on \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\), as an \(n\)-fold tensor power of a functor in the left class, the functor \(\mathcal X^{\otimes n} \rightarrow\mathcal Y^{\otimes n}\) remains surjective on objects and dominant on \(1\)-morphisms, and hence the first map is an equivalence as a direct consequence of the orthogonality of surjective-on-objects-and-dominant-on-1-morphisms and faithful functors. ◻