Corollary 7.4.1.12. Let $n$ be an integer, let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ be a diagram. Suppose that, for every vertex $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\mathscr {F}(C)$ is locally $n$-truncated. Then the limit $\varprojlim ( \mathscr {F} )$ is a locally $n$-truncated $\infty $-category.
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Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is an $\infty $-category and that $n \geq -2$. Let $\operatorname{\mathcal{E}}= \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ denote the $\infty $-category of elements of $\mathscr {F}$. It follows from Variant 5.1.5.17 that the projection map $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is an essentially $(n+1)$-categorical cocartesian fibration. Applying Corollary 4.8.6.21, we see that the $\infty $-category of sections $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ is locally $n$-truncated. Since $\varprojlim (\mathscr {F} )$ can be identified with a full subcategory of $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$ (Corollary 7.4.1.10), it is also locally $n$-truncated (Remark 4.8.2.3). $\square$