ScalingStacks

Proof. For $\mathcal{C}=\mathcal{S}$, this follows from inspecting the induced map between the long exact sequences of homotopy groups associated with the vertical maps. For $\mathcal{C}=\mathcal{S}^{K}$, this follows from the claim for $\mathcal{S}$, since both truncation and pullbacks are computed level-wise. A general $\infty$-topos is a left exact localization of $\mathcal{S}^{K}$ for some $K$, and left exact colimit-preserving functors between presentable $\infty$-categories commute with truncation by T.5.5.6.28 and with pullbacks by assumption. Finally, by T.6.4.1.5 every $m$-topos is the full subcategory on $\left(m-1\right)$-truncated objects in an $\infty$-topos and this full subcategory is closed under limits. ∎