ScalingStacks

Consider an adjunction between symmetric monoidal \(\infty\)-categories with (strongly) symmetric monoidal left adjoint \(L\) and denote the induced adjunction between \(\infty\)-categories of \(\mathbb E_1\)-algebras by Then, for any morphism of \(\mathbb E_1\)-algebras \(f \colon L_{\mathbb E_1} a \rightarrow b\) in \(\mathcal W\), the induced map of spaces \[\mathrm{PreBraid}_{\mathcal W}(f) \rightarrow\mathrm{PreBraid}_{\mathcal V}(R_{\mathbb E_1}f) \rightarrow\mathrm{PreBraid}_{\mathcal V}( R_{\mathbb E_1}f \circ \eta_a)\] (constructed as in lemma 7.2.9) is an equivalence, where \(\eta\) denotes the unit of the adjunction.