Let \(\mathcal C\in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\), \(\mathcal D\in \mathrm{Alg}_{\mathbb E_{\infty}}(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}])\) and let \(H \colon \mathcal C\rightarrow\mathcal D\) be a morphism in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}[\mathrm{add}_{k}^{B\mathbb{Z}}])\) whose underlying \((\infty,2)\)-functor is faithful, i.e. induces fully faithful functors on hom-categories. Consider the monoidal functor \({\mathbf K}^b_{\mathrm{loc}}(\mathcal C) \rightarrow\mathcal D\), induced by the adjunction ([00D4]), as an object of \(\mathrm{Alg}_{\mathbb E_1}\left(\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}\right)\).
Then, the map of spaces (constructed more formally in the proof below) \[ \mathrm{Braid}_{\mathrm{Cat}[\mathrm{st}^{B\mathbb{Z}}_{k}]_{/\mathcal D}}({\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal C\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)),\] which restricts a braiding on \({\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) to a prebraiding on the subcategory inclusion \(\mathcal C\rightarrow{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)\) and then passes to the homotopy \(1\)-category \(h_1\), is an equivalence. (Here, we leave the evident maps to \(\mathcal D\) and \(h_1\mathcal D\) implicit.)
Further, assume there is a functor \(\iota \colon \mathcal B\rightarrow\mathcal C\) in \(\mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{(\infty,2)})\) which is surjective on objects and such that for every two objects \(b,b' \in \mathcal B\), any object in \(\underline{\mathrm{Hom}}_{\mathcal C}(\iota b, \iota b') \in \mathrm{add}_{k}^{B\mathbb{Z}}\) is a retract of a finite coproduct of \(\mathbb{Z}\)-shifts of objects in the image of \(\underline{\mathrm{Hom}}_{\mathcal B}(b, b') \in \mathrm{Cat}_{(\infty,1)}\). Then, the pre-composition map \[\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal C\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C)) \rightarrow\mathrm{PreBraid}_{{\mathrm{Cat}_{({1}, {1})}}_{/h_1\mathcal D}}(h_1 \mathcal B\rightarrow h_1{\mathbf K}^b_{\mathrm{loc}}(\mathcal C))\] is an equivalence of spaces.