For an ordinary monoidal \(1\)-category \(\mathcal A\in \mathrm{Alg}_{\mathbb E_1}(\mathrm{Cat}_{({1}, {1})})\) it follows from corollary 7.7.8 that the map of spaces \[\mathrm{Braid}_{\mathrm{Cat}_{({1}, {1})}}(\mathcal A) \rightarrow\mathrm{PreBraid}_{\mathrm{Cat}_{({1}, {1})}}(\mathcal A).\] is an equivalence. By corollary 7.4.15([00FC]), these spaces are equivalent to the (discrete) set of classical braidings on \(\mathcal A\).
Similarly, it follows from corollary 7.4.15([00FB]) that for monoidal functors \(F \colon \mathcal A\rightarrow\mathcal B\) between ordinary monoidal \(1\)-categories, the spaces \(\mathrm{PreBraid}_{\mathrm{Cat}_{({1}, {1})}}(F)\) are equivalent to the (discrete) set of classical prebraidings on \(F\) in the sense of definition 2.4.8.