Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Warning 5.1.6.6. Assertion $(2)$ of Proposition 5.1.6.5 need not be true if $U$ and $V$ are only assumed to be inner fibrations. For example, let $\operatorname{\mathcal{E}}$ be an $\infty $-category and let $\operatorname{\mathcal{E}}' \subseteq \operatorname{\mathcal{E}}$ be a full subcategory for which the inclusion map $\iota : \operatorname{\mathcal{E}}' \hookrightarrow \operatorname{\mathcal{E}}$ is an equivalence. Then we have a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}' \ar [rr]^{\iota } \ar [dr]_{\iota } & & \operatorname{\mathcal{E}}\ar [dl]^{\operatorname{id}} \\ & \operatorname{\mathcal{E}}, & } \]

where the vertical maps are inner fibrations. However, $\iota $ is not an equivalence of inner fibrations over $\operatorname{\mathcal{E}}$ unless $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{E}}$.