Let \(n \geq 0\) and consider categories and functors in \(\mathrm{Cat}_{(\infty,2)}\) 
with faithfulness properties as indicated. Then, the map induced by applying \(h_1\) \[\mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/\mathcal W}}\left(\mathcal Y^{\times n}, \mathcal Z\right) \rightarrow\mathrm{Hom}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal W}}\left(h_1 \mathcal Y^{\times n} , h_1 \mathcal Z\right)\] is an equivalence of spaces.
Proof.
Since \(h_1 \colon \mathrm{Cat}_{(\infty, {2})}\rightarrow\mathrm{Cat}_{({1}, {1})}\) is (strongly) symmetric monoidal, this follows directly from corollary 5.5.7. ◻
Original source: arXiv:2401.02956v2
Original source · 2401.02956v2