Proof.
The pair of composable morphisms of \(\mathbb E_1\)-algebras \(\{f,g\}\) may be corepresented by an operad map \([2] \otimes \mathbb E_1 \rightarrow\mathcal O\). Recall the operad maps \([2] \otimes \mathbb E_0 \rightarrow\mathbb T_2 \sqcup_{\{0<2\}} [2] \rightarrow\mathbb T_2 \sqcup_{\{1<2\}}\) from observation 7.2.10 corepresenting a pair of composable \(\mathbb E_0\)-morphisms with a \(\mathbb T_2\)-structure on \(g\) and on \(g\circ f\), respectively, and the construction of a \(\mathbb T_2\)-structure on \(g\) from a \(\mathbb T_2\)-structure on \(g\circ f\).
Fix a prebraiding on \(g\circ f\), corepresented by a lift of the operad map \([2] \otimes \mathbb E_1 \rightarrow\mathcal O\) to an operad map \(\mathbb T_2 \sqcup_{\{0<2\}}[2] \otimes \mathbb E_1 \rightarrow\mathcal O\). The fiber of \(\mathrm{PreBraid}_{\mathcal O}(g) \rightarrow\mathrm{PreBraid}_{\mathcal O}(g\circ f)\) is given by the space of lifts 
We now claim that both operad maps \[
[2] \otimes \mathbb E_1 \rightarrow\left(\mathbb T_2 \sqcup_{\{0<2\}}{[2]} \right) \otimes \mathbb E_1 \qquad [2] \otimes \mathbb E_1 \rightarrow\left( \mathbb T_2 \sqcup_{\{1<2\}}{[2]} \right) \otimes \mathbb E_1\] are \(0\)-surjective. Indeed, this follows since \([1] \otimes \mathbb E_1 \rightarrow\mathbb T_2 \otimes \mathbb E_1\) is \(0\)-surjective by proposition 7.6.1 and since both operad maps ([00HN]) are by definition given by a pushout of this \(0\)-surjective operad map against the operad maps \([1] \otimes \mathbb E_1 \rightarrow[2] \otimes \mathbb E_1\) induced by the inclusions \(\{0<2\} \rightarrow[2]\) and \(\{1<2\} \rightarrow[2]\), respectively (see observation 7.2.10).
Hence, it follows from lemma 8.3.3 applied to the \(\infty\)-category \(\mathrm{Op}\) that our space of lifts is equivalent to the space of lifts 
Therefore, without loss of generality, we may replace \(\mathcal O\) by \(\mathcal O|_{[2] \otimes \mathbb E_1}\) and hence with observation 8.3.2, we may assume that \[\mathrm{Mul}_{\mathcal O}(b, \ldots, b; c) \rightarrow\mathrm{Mul}_{\mathcal O}(a, \ldots, a; c)\] are equivalences. It therefore follows from proposition 7.8.2.([00GI]) that \[\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}(\mathcal O)}(b, \ldots, b; c) \rightarrow\mathrm{Mul}_{\mathrm{Alg}_{\mathbb E_1}{\mathcal O}}(a, \ldots, a; c)\] are equivalences, and hence as in the proof of corollary 8.3.5 that \[\mathrm{PreBraid}_{\mathcal O}(g) \rightarrow\mathrm{PreBraid}_{\mathcal O}(g\circ f)\] are equivalences. ◻