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Theorem 5.6.3.6 (Universality Theorem). Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$. For every simplicial set $\operatorname{\mathcal{C}}$, the construction

\[ ( \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{Q}}) \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F} \]

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

Proof of Theorem 5.6.3.6. Let $\operatorname{\mathcal{Q}}$ be a full subcategory of $\operatorname{\mathcal{QC}}$ and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a $\operatorname{\mathcal{Q}}$-small cocartesian fibration. We wish to show that $U$ admits a covariant transport representation $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{Q}}$, which is uniquely determined up to isomorphism (as an object of the functor $\infty $-category $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{Q}})$). The existence statement follows from Corollary 5.6.4.11, and the uniqueness from Corollary 5.6.4.12. $\square$