Let \(\mathcal O, \mathcal P\) be \(\infty\)-operads and let \(b\) and \(c\) be \(\mathcal O\)-algebras in \(\mathcal P\).
Let \(F \colon \mathcal P\rightarrow\mathcal Q\) be an operad map such that for all \(n\geq 0\) and colors \(X_1,\ldots, X_n, Y\) in \(\mathcal O\), the map of spaces \[\mathrm{Mul}_{\mathcal P}(b_{X_1}, \ldots, b_{X_{n}}; c_{Y}) \xrightarrow{F(-)} \mathrm{Mul}_{\mathcal Q}(Fb_{X_1}, \ldots, Fb_{X_{n}}; Fc_{Y})\] is an equivalence. Then, for any \(m \geq 0\), the map of multi-hom spaces \[\mathrm{Mul}_{\mathrm{Alg}_{\mathcal O}(\mathcal P)}(\underbrace{b,\ldots, b}_{m}; c) \xrightarrow{F(-)} \mathrm{Mul}_{\mathrm{Alg}_{\mathcal O}(\mathcal Q)}(\underbrace{Fb,\ldots, Fb}_{m}; Fc)\] is an equivalence.
Let \(f \colon a \rightarrow b\) be a morphism of \(\mathcal O\)-algebras in \(\mathcal P\). Assume that for all \(n \geq 0\) and colors \(X_1, \ldots, X_n, Y\) in \(\mathcal O\), the map of spaces \[\mathrm{Mul}_{\mathcal P}(b_{X_1}, \ldots, b_{X_{n}}; c_{Y}) \xrightarrow{-\circ(f,\ldots, f)} \mathrm{Mul}_{\mathcal P}(a_{X_1}, \ldots, a_{X_{n}}; c_{Y})\] is an equivalence. Then, for any \(m \geq 0\), the map of multi-hom spaces \[\mathrm{Mul}_{\mathrm{Alg}_{\mathcal O}(\mathcal P)}(\underbrace{b,\ldots, b}_m; c) \xrightarrow{- \circ (f,\ldots, f)} \mathrm{Mul}_{\mathrm{Alg}_{\mathcal O}(\mathcal P)}(\underbrace{a, \ldots, a}_m; c)\] is an equivalence for all \(n\).