ScalingStacks

Proof. (2) Assume that $d\geq 1$. By the analogous fact for $\infty$-categories, the composition with $\theta_{d}$ induces an isomorphism

$$\operatorname{Fun}_{\mathbf{Fin}_{*}}((h_{d}\mathcal{O})^{\otimes},\mathcal{U}^{\otimes})\overset{\sim}{\longrightarrow}\operatorname{Fun}_{\mathbf{Fin}_{*}}(\mathcal{O}^{\otimes},\mathcal{U}^{\otimes}).$$

The simplicial set $\underline{\operatorname{Alg}}_{\mathcal{O}}\left(\mathcal{U}\right)$ is the full subcategory of $\operatorname{Fun}_{\mathbf{Fin}_{*}}\left(\mathcal{O}^{\otimes},\mathcal{U}^{\otimes}\right)$ spanned by maps of $\infty$-operads (and similarly for $h_{d}\mathcal{O}$ instead of $\mathcal{O}$). The claim now follows from the fact that the image of a coCartesian edge in $\mathcal{O}^{\otimes}$ is coCartesian in $\left(h_{d}\mathcal{O}\right)^{\otimes}$ and, conversely, every inert morphism in $\left(h_{d}\mathcal{O}\right)^{\otimes}$ is up to equivalence the image of an inert morphism in $\mathcal{O}^{\otimes}$ (lift the source to some object $X\in\mathcal{O}^{\otimes}$ and choose any inert map with domain $X$).

For $d=0$, essentially the same argument works, only now the inert maps of $\left(h_{0}\mathcal{O}\right)^{\otimes}$ are precisely those whose image in $\mathbf{Fin}_{*}$ is inert and therefore the inert maps of $\left(h_{0}\mathcal{O}\right)^{\otimes}$ are again precisely the images of inert maps in $\mathcal{O}^{\otimes}$. For $d=-1$, the claim is obvious.