Corollary 4.8.3.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $n \geq -2$ be an integer. Then $\operatorname{\mathcal{C}}$ is locally $n$-truncated if and only if the restriction map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{m}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{m}, \operatorname{\mathcal{C}}) \]
is surjective for every integer $m \geq n+3$.