ScalingStacks

Proof. The map $\theta_{d}\colon\mathcal{O}\to h_{d}\mathcal{O}$ is surjective on objects and hence is essentially surjective. For $d\geq 1$, the second assertion follows from the corresponding fact for $\infty$-categories; and for $d=-1,0$, it follows directly from the definition. ∎