Corollary 2.3.1.10. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then the restriction map
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n}, \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^{n}, \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}) ) \]
is bijective for $n \geq 4$ and injective when $n=3$.