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Corollary 5.6.3.12 (The Universal Cocartesian Fibration). For every simplicial set $\operatorname{\mathcal{C}}$, the construction

\[ \mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F} = \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{QC}}} \operatorname{\mathcal{QC}}_{\operatorname{Obj}} \]

induces a bijection from $\pi _0( \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})^{\simeq } )$ to the collection of equivalence classes of cocartesian fibrations $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ having essentially small fibers.

Proof. Apply Theorem 5.6.3.6 in the case $\operatorname{\mathcal{Q}}= \operatorname{\mathcal{QC}}$. $\square$