Let \(n \geq 0\) and consider categories and functors in \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\) 
with surjectivity and faithfulness properties as indicated (and where \(\otimes\) denotes the tensor product in \(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]\)). Then, the map induced by precomposition with the tensor power \(\mathcal X^{\otimes n} \rightarrow\mathcal Y^{\otimes n}\) \[\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal W}}\left(\mathcal Y^{\otimes n}, \mathcal Z\right) \rightarrow\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal W}}\left( \mathcal X^{\otimes n} , \mathcal Z\right)\] is an equivalence of spaces.
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. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2