Kerodon

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

Corollary 4.5.2.25. Suppose we are given a diagram of $\infty $-categories

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

where $U'$ and $U$ are isofibrations and the functors $F$ and $F'$ are equivalences of $\infty $-categories. Then, for every object $D' \in \operatorname{\mathcal{D}}'$ having image $D = F(D') \in \operatorname{\mathcal{D}}$, the induced functor

\[ \operatorname{\mathcal{C}}'_{D'} = \{ D'\} \times _{\operatorname{\mathcal{D}}'} \operatorname{\mathcal{C}}' \rightarrow \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}_{D} \]

is an equivalence of $\infty $-categories.