Corollary 5.6.0.6. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the construction
\[ ( \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}) \mapsto (\int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}) \]
induces a bijection from $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}})^{\simeq } )$ to the set of equivalence classes of essentially small left fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$.