Proof.
Recall corollary 5.5.5, that for any small monoidal \((\infty,2)\)-category \(\mathcal D\in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{(\infty, {2})})\), taking the homotopy 1-category induces an equivalence \[h_1 \colon {\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{(\infty, {2})})}_{\small{/^{\tiny{\text{f}}\,}}{\mathcal D}} \rightarrow{\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{({1}, {1})})}_{\small{/^{\tiny{\text{f}}\,}}{h_1\mathcal D}}\] between the full subcategories of \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{(\infty, {2})})_{/\mathcal D}\) and \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{({1}, {1})})_{/h_1\mathcal D}\) on the faithful functors. Applying this to \(\mathcal D= \mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\), the monoidal \((2,2)\)-category \(\mathrm{BSbim}\) is the unique pre-image of \(h_1 \mathrm{BSbim}\rightarrow h_1 \mathrm{Mor}^{\mathrm{poly}, \mathrm{gr-proj}}(\mathrm{mod}_k^{\mathbb{Z}})\). Since any \((\infty,2)\)-category with a faithful functor to a \((2,2)\)-category is again a \((2,2)\)-category, it follows that \(\mathrm{BSbim}\) is a \((2,2)\)-category. ◻