$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 5.1.7.10. Suppose we are given a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\ar [rr]^-{F} \ar [dr]_{U} & & \operatorname{\mathcal{D}}\ar [dl]^{V} \\ & \operatorname{\mathcal{E}}, & } \]
where $U$ and $V$ are inner fibrations. Then $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ if and only if it satisfies the following pair of conditions:
- $(1)$
The functor $F$ is fully faithful.
- $(2)$
For every object $E \in \operatorname{\mathcal{E}}$, the functor $F_{E}: \operatorname{\mathcal{C}}_{E} \rightarrow \operatorname{\mathcal{D}}_{E}$ is an equivalence of $\infty $-categories.
Proof.
If $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$, then it is an equivalence of $\infty $-categories (Proposition 5.1.7.5) and each of the functors $F_{E}$ has the same property (Remark 5.1.7.4); this proves the necessity of conditions $(1)$ and $(2)$. Conversely, suppose that $F$ satisfies conditions $(1)$ and $(2)$; we will show that $F$ is an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$. By virtue of Proposition 5.1.7.9, it will suffice to show that for every morphism $\sigma : \Delta ^{n} \rightarrow \operatorname{\mathcal{E}}$, the induced map $F_{\sigma }: \Delta ^ n \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^{n} \times _{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ is an equivalence of $\infty $-categories. It follows from assumption $(1)$ and Variant 4.8.6.19 that the functor $F_{\sigma }$ is fully faithful, and from assumption $(2)$ that the functor $F_{\sigma }$ is essentially surjective. The desired result now follows from Theorem 4.6.2.21.
$\square$