Proof of theorem 8.2.1.
Given categories and functors as in part ([00H4]) of theorem 8.2.1, we will prove that the composite \[\begin{aligned} \mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}({\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) &\rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal C\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \\ \nonumber &\rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal B\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \end{aligned}\] is an equivalence. Note that part ([00H2]) of theorem 8.2.1 then follows by taking \(\mathcal B\rightarrow\mathcal C\) to be the identity \(\mathcal C\rightarrow\mathcal C\) (which clearly satisfies the required conditions). Then, the second statement follows since the first map and the composite in ([00I7]) are equivalences, and hence so is the second map.
To prove that ([00I7]) is an equivalence, note that it follows from lemma 6.3.1 that the condition on \(\mathcal B\rightarrow\mathcal C\) in the statement of theorem 8.2.1.([00H4]) equivalently asserts that \(\mathrm{Lin}_{k, \mathrm{loc}}^{\mathbb{Z}} (\mathcal B) \rightarrow\mathcal C\) is surjective on objects and dominant on \(1\)-morphisms.
We now unpack ([00I7]) as a sequence of equivalences of spaces of (pre-)braidings:
Diagram references: 8.3.4 8.4.4 ([00I2]) 8.3.7 8.3.6 8.4.4 ([00I3]) 8.3.7 8.3.5 8.4.4 ([00I4]) This completes the proof of theorem 8.2.1. ◻