Kerodon

Proof. We will give the proof under the assumption that $U$ is a cocartesian fibration; the proof when $U$ is a cartesian fibration is similar. Let $S$ be the collection of all monomorphisms of simplicial sets $f: A \hookrightarrow B$ with the following property: for every morphism of simplicial sets $B \rightarrow \operatorname{\mathcal{C}}$, the induced map $A \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}\hookrightarrow B \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a categorical equivalence. To complete the proof, it will suffice to show that the morphism $\overline{F}: \operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ belongs to $S$. In fact, we claim that every inner anodyne morphism of simplicial sets belongs to $S$. Using Remark 4.5.3.6, Remark 4.5.3.5, Corollary 4.5.7.2, and Remark 4.5.4.13, we see that $S$ is weakly saturated (see Definition 1.5.4.12). It will therefore suffice to show that $S$ contains every inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$, $0 < i < n$. In particular, we are reduced to proving Lemma 5.3.6.5 in the special case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ is the nerve of a category $\operatorname{\mathcal{C}}_0$. Applying Corollary 5.3.4.9, we deduce that there exists a diagram of $\infty $-categories $\mathscr {G}: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{QCat}$ and a scaffold $\lambda : \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \rightarrow \operatorname{\mathcal{E}}$. We then have a commutative diagram of simplicial sets

where the vertical maps are categorical equivalences (Lemma 5.3.6.4). Consequently, to show that $F$ is a categorical equivalence, it will suffice to show that $\widetilde{F}$ is a categorical equivalence, which follows from Lemma 5.3.6.3. $\square$