$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Theorem 7.4.2.4. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the construction
\[ ( \mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}) \mapsto ( U: \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}) \]
determines an equivalence of homotopy categories
\[ \mathrm{h} \mathit{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) } \rightarrow \mathrm{h} \mathit{ \operatorname{LFib}( \operatorname{\mathcal{C}}) }, \]
which carries (the homotopy class of) a natural transformation $\gamma : \mathscr {F} \rightarrow \mathscr {F}'$ to (the homotopy class) of a map $\Gamma \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \int _{\operatorname{\mathcal{C}}} \mathscr {F}, \int _{\operatorname{\mathcal{C}}} \mathscr {F}' )$ which is compatible with $\gamma $.
Proof of Theorem 7.4.2.4.
Let $\operatorname{\mathcal{C}}$ be a simplicial set. It follows from Proposition 7.4.2.6 (together with Example 7.4.2.9 and Remark 7.4.2.10) that there is a unique functor
\[ \Theta : \mathrm{h} \mathit{ \operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}) } \rightarrow \mathrm{h} \mathit{ \operatorname{LFib}(\operatorname{\mathcal{C}}) } \]
which is given on objects by the construction $\mathscr {F} \mapsto \int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and on morphisms by the construction $[ \gamma ] \mapsto [\Gamma ]$, where $\gamma : \mathscr {F} \rightarrow \mathscr {F}'$ denotes a natural transformation and $\Gamma \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \int _{\operatorname{\mathcal{C}}} \mathscr {F}, \int _{\operatorname{\mathcal{C}}} \mathscr {F}' )$ is chosen to be compatible with $\gamma $. It follows from Corollary 5.6.0.6 that the functor $\Theta $ is bijective on isomorphism classes: in particular, it is essentially surjective. To complete the proof, it suffices to show that $\Theta $ is fully faithful, which is a reformulation of Proposition 7.4.2.11.
$\square$