Corollary 10.3.1.13. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\operatorname{\mathcal{D}}= \mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category. Then the construction $(\operatorname{\mathcal{D}}^{0} \subseteq \operatorname{\mathcal{D}}) \mapsto \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}^{0} ) \times _{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}}) } \operatorname{\mathcal{C}}$ induces a bijection
\[ \{ \textnormal{Sieves $\operatorname{\mathcal{D}}^{0} \subseteq \operatorname{\mathcal{D}}$} \} \rightarrow \{ \textnormal{Sieves $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$} \} . \]