Corollary 9.3.1.25. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n$ be an integer. Then:
If $\operatorname{\mathcal{C}}$ is an $n$-truncated object of $\operatorname{\mathcal{QC}}$ (in the sense of Definition 9.3.1.1), then it is locally $n$-truncated (in the sense of Definition 4.8.2.1).
If $n \geq -1$ and $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated, then it is an $n$-truncated object of $\operatorname{\mathcal{QC}}$.