$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 9.4.6.20. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [rr]^-{F} \ar [dr]_{U} & & \operatorname{\mathcal{E}}' \ar [dl]^{ U' } \\ & \operatorname{\mathcal{C}}, & } \]
where $U$ is a flat inner fibration, $U'$ is an inner fibration, and $F$ is an equivalence of $\infty $-categories. Then:
- $(1)$
For every object $C \in \operatorname{\mathcal{C}}$, the induced map of fibers $F_{C}: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ is a Morita equivalence.
- $(2)$
The inner fibration $U'$ is flat.
Proof of Lemma 9.4.6.20.
We will prove $(1)$; assertion $(2)$ then follows from Example 9.4.6.10. Using Corollary 4.5.2.23, we can factor the inclusion map $\{ C\} \hookrightarrow \operatorname{\mathcal{C}}$ as a composition $\{ C\} \hookrightarrow \widetilde{\operatorname{\mathcal{C}}} \xrightarrow {V} \operatorname{\mathcal{C}}$, where $V$ is an isofibration and $\widetilde{\operatorname{\mathcal{C}}}$ is a contractible Kan complex. Set $\widetilde{\operatorname{\mathcal{E}}} = \widetilde{\operatorname{\mathcal{C}}} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ and $\widetilde{\operatorname{\mathcal{E}}}' = \widetilde{\operatorname{\mathcal{C}}} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}'$, so that we have a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}_{C} \ar [r]^-{F_{C}} \ar [d] & \operatorname{\mathcal{E}}'_{C} \ar [d] \\ \widetilde{\operatorname{\mathcal{E}}} \ar [r]^-{\widetilde{F}} & \widetilde{\operatorname{\mathcal{E}}}'. } \]
It follows from Theorem 9.4.6.6 that the vertical maps are Morita equivalences, and from Corollary 4.5.2.29 that $\widetilde{F}$ is an equivalence of $\infty $-categories. Applying Remark 9.4.5.5, we conclude that $F_{C}$ is a Morita equivalence.
$\square$