Given categories and functors as in the assumptions of theorem 8.2.1. Then, for each \(n \geq 0\), the following hold:
The space \[\mathrm{Im}\left( S_n \rightarrow\mathrm{Hom}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}\left( {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \right)\] is \(1\)-truncated, i.e. a \(1\)-groupoid.
The map of spaces \[\begin{aligned} \mathrm{Im}\left(S_n \rightarrow\vphantom{\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}}}\right. & \left. \mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}} \left( \mathcal C^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\right)\right) \\ &\longrightarrow \mathrm{Im}\left(S_n \rightarrow\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}}\left(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathcal B)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \right) \end{aligned}\] is an equivalence.
The map of spaces \[\begin{aligned} \mathrm{Im}\left( S_n \rightarrow\vphantom{\mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/\mathcal D}}} \right. & \left. \mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/\mathcal D}} \left( \mathcal B^{\times n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\right) \right) \\ &\longrightarrow \mathrm{Im}\left( S_n \rightarrow\mathrm{Hom}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}\left(h_1\mathcal B^{\times n}, h_1 {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \right) \end{aligned}\] is an equivalence