Recall the (\(0\)-surjective, \(0\)-faithful) factorization system on the \(\infty\)-category \(\mathrm{Op}\) from definition 7.5.1 and proposition 7.5.3.
Given a map of operads \(\mathbb E_1 \rightarrow\mathcal O\), corepresenting an \(\mathbb E_1\)-algebra \(A\) in \(\mathcal O\), we write \(\mathcal O|_{\mathbb E_1}\) for the factorization \(\mathbb E_1 \rightarrow\mathcal O|_{ \mathbb E_1} \rightarrow\mathcal O\) into a \(0\)-surjective followed by a \(0\)-faithful map of operads.
Explicitly, \(\mathcal O|_{\mathbb E_1}\) has one color \(A\) and the only non-empty multi-hom spaces are given by the full images \[\mathrm{Mul}_{\mathcal O|_{\mathbb E_1}}(A, \ldots, A; A) = \mathrm{Im}\left( S_n \rightarrow\mathrm{Mul}_{\mathcal O}(A, \ldots, A; A) \right)\] of the map \(\mathbb E_1(n) = S_n \rightarrow\mathrm{Mul}_{\mathcal O}(A, \ldots, A ; A)\) induced by the \(\mathbb E_1\)-structure on \(A\).
(In other words, \(\mathrm{Im}\left( S_n \rightarrow\mathrm{Mul}_{\mathcal O}(A, \ldots, A; A) \right)\) is precisely the union of those components of \(\mathrm{Mul}_{\mathcal O}(A, \ldots, A ; A)\) which contain the orbit of the \(n\)-ary multiplication of \(A\) under the \(S_n\)-action permuting its inputs.)
Given a map of operads \([1] \otimes \mathbb E_1 \rightarrow\mathcal O\), corepresenting a morphism \(f \colon A \rightarrow B\) of \(\mathbb E_1\)-algebras in \(\mathcal O\), we write \(\mathcal O|_{[1]\otimes \mathbb E_1}\) for the factorization \([1] \otimes \mathbb E_1 \rightarrow\mathcal O|_{[1] \otimes \mathbb E_1} \rightarrow\mathcal O\) into a \(0\)-surjective followed by a \(0\)-faithful map of operads.
Explicitly, \(\mathcal O|_{[1]\otimes \mathbb E_1}\) has (at most) two colors \(A, B\) and multi-hom spaces connecting them, one of them being \[\mathrm{Mul}_{\mathcal O|_{[1] \otimes \mathbb E_1}}(A, \ldots, A; B) = \mathrm{Im}\left(S_n \rightarrow\mathrm{Mul}_{\mathcal O}(A, \ldots, A; B)\right),\]where the map from \(S_n = \mathbb E_1(n)\) is induced by the \(\mathbb E_1\)-structures on \(f\). (In other words, \(\mathrm{Im}\left( S_n \rightarrow\mathrm{Mul}_{\mathcal O}(A, \ldots, A; B) \right)\) is the union of those components of \(\mathrm{Mul}_{\mathcal O}(A, \ldots, A ; B)\) which contain the orbit of \(f \circ \mu_A \simeq \mu_B \circ f\) under the \(S_n\)-action permuting its inputs.)
Given a map of operads \([2] \otimes \mathbb E_1 \rightarrow\mathcal O\), corepresenting a composable pair of \(\mathbb E_1\)-algebra morphisms \(A \rightarrow B \rightarrow C\), we can similarly consider \(\mathcal O|_{[2] \otimes \mathbb E_1}\), which has (at most) three objects \(A, B, C\), and multi-hom spaces connecting them, such as \[\begin{aligned} \mathrm{Mul}_{\mathcal O|_{[2] \otimes \mathbb E_1}}(A, \ldots, A; B)& = \mathrm{Im}\left(S_n \rightarrow\mathrm{Mul}_{\mathcal O}(A, \ldots, A; B)\right)\\ \mathrm{Mul}_{\mathcal O|_{[2] \otimes \mathbb E_1}}(B, \ldots, B; C) &= \mathrm{Im}\left(S_n \rightarrow\mathrm{Mul}_{\mathcal O}(B, \ldots, B; C)\right), \end{aligned}\] where the maps from \(S_n = \mathbb E_1(n)\) are induced by the \(\mathbb E_1\)-structure on \(f\) and \(g\), respectively.