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Lemma 5.3.6.5. Suppose we are given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [d] \ar [r]^-{F} & \operatorname{\mathcal{E}}\ar [d]^-{U} \\ \operatorname{\mathcal{C}}' \ar [r]^-{\overline{F}} & \operatorname{\mathcal{C}}, } \]

where $\overline{F}$ is inner anodyne. If $U$ is either a cartesian fibration or a cocartesian fibration, then $F$ is a categorical equivalence of simplicial sets.

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 1.4.4.15). 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

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \ar [r]^{ \widetilde{F} } \ar [d] & \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {G} ) \ar [d]^{\lambda } \\ \operatorname{\mathcal{E}}' \ar [r]^-{F} & \operatorname{\mathcal{E}}, } \]

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. $\square$