ScalingStacks

Proof. Let $g\colon\mathcal{P}\to\mathcal{Q}$ be a map of reduced $\infty$-operads such that $g_{\mathbf{SSeq}}$ is an equivalence. The map $g$ is defined by a commutative triangle

To show that $g$ is an equivalence of $\infty$-operads, we need to show that $g^{\otimes}$ is an equivalence of $\infty$-categories. Since $\mathcal{P}$ and $\mathcal{Q}$ are reduced, it is clear that $g^{\otimes}$ is essentially surjective. To show that $g^{\otimes}$ is fully faithful, we can use the Segal conditions to reduce this to showing that the map

$$\mathcal{P}\left(n\right)=\operatorname{Mul}_{\mathcal{P}}(*^{(n)},*)\to\operatorname{Mul}_{\mathcal{Q}}(*^{(n)},*)=\mathcal{Q}\left(n\right)$$

is a homotopy equivalence for all $n$. By 2.3.5, those maps are induced by the equivalence $g_{\mathbf{SSeq}}$ and therefore are equivalences. ∎