Kerodon

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

Corollary 4.5.2.18. Suppose we are given a commutative diagram of $\infty $-categories

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d] & \operatorname{\mathcal{C}}_1 \ar [l] \ar [d] \\ \operatorname{\mathcal{D}}_0 \ar [r] & \operatorname{\mathcal{D}}& \operatorname{\mathcal{D}}\ar [l] } \]

where the vertical maps are equivalences of $\infty $-categories. Then the induced map $\operatorname{\mathcal{C}}_{0} \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{D}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}_1$ is an equivalence of $\infty $-categories.