Proof.
We first prove the part ([00I3]). Since the monoidal structure on \(\mathcal C\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) arises from adjunction, the relevant maps from \(S_n\) all factor as: 
Since \(\mathcal C\rightarrow\mathcal D\) is faithful, it follows from proposition 8.4.2 that the top horizontal map is an equivalence. Moreover, for every \(f\in \mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}^{B \mathbb{Z}}_k]_{/\mathcal D}}(\mathcal C^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C))\), the induced map between the fibers of the vertical maps is \[\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}^{B \mathbb{Z}}_k]_{/{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)}}(\mathcal C^{\otimes n}, \mathcal C) \rightarrow\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)}}\left(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathcal B)^{\otimes n}, \mathcal C\right).\] It follows from proposition 4.3.2 that \(\mathcal C\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) is faithful. Hence, this map between fibers is also an equivalence by proposition 8.4.2. Therefore, it follows from lemma 8.4.3 that the induced map between the full images of the vertical maps is an equivalence, and hence so is also the induced map between the full images of \(S_n\).
The proof of part ([00I3]) is entirely analogous: The relevant maps from \(S_n\) all factor as 
Since \(\mathcal C\rightarrow\mathcal D\) is faithful by assumption, it follows from corollary 8.4.1 that the top horizontal map is an equivalence. The induced map between the fibers of the vertical maps at an \(f\in \mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/\mathcal D}}(\mathcal B^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C))\) is given by \[\mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)}}(\mathcal B^{\otimes n}, \mathcal C) \rightarrow\mathrm{Hom}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)}}\left(h_1 \mathcal B^{\times n}, h_1\mathcal C\right)\] and hence is also an equivalence by corollary 8.4.1 since also \(\mathcal C\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) is faithful. Therefore, it follows from lemma 8.4.3 that the induced map between the full images of the vertical maps is an equivalence, and hence so is also the induced map between the full images of \(S_n\).
Part ([00I2]) follows from combining parts ([00I3]) and ([00I4]): It follows from adjunction and monoidality of \({\mathbf K}^b_{\mathrm{loc}}(-)\) that \[\mathrm{Hom}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}\left( {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \simeq \mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}}\left( \mathcal C^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\right).\] Similarly, it follows from adjunction and monoidality of \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (-)\) that \[\mathrm{Hom}_{\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}]_{/\mathcal D}}\left( \mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathcal B)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \simeq \mathrm{Hom}_{{\mathrm{Cat}_{(\infty, {2})}}_{/\mathcal D}} \left( \mathcal B^{\times n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\right).\] Hence, combining the second and third statement, we find that \[\mathrm{Im}\left( S_n \rightarrow\mathrm{Hom}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}\left( {\mathbf K}^b_{\mathrm{loc}}(\mathcal C)^{\otimes n}, {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \right) \simeq \mathrm{Im}\left( S_n \rightarrow\mathrm{Hom}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}\left(h_1\mathcal B^{\times n}, h_1 {\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \right) \right)\] which is — as a mapping space of \(\mathrm{Cat}_{({1}, {1})}\) — a \(1\)-groupoid. ◻