Notation 7.4.2.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set. We let $\operatorname{LFib}( \operatorname{\mathcal{C}})$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{ / \operatorname{\mathcal{C}}}$ spanned by the (essentially small) left fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. We will regard $\operatorname{LFib}( \operatorname{\mathcal{C}})$ as a simplicially enriched category, with morphism spaces given by
\[ \operatorname{Hom}_{ \operatorname{LFib}(\operatorname{\mathcal{C}}) }( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' )_{\bullet } = \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}, \operatorname{\mathcal{E}}' ). \]
It follows from Corollary 4.4.2.5 that the simplicial category $\operatorname{LFib}(\operatorname{\mathcal{C}})$ is locally Kan.