Kerodon

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

Proposition 4.5.2.19. Suppose we are given a commutative diagram of $\infty $-categories

4.25
\begin{equation} \begin{gathered}\label{equation:categorical-pullback-square3} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}' \ar [r] \ar [d]^{F'} & \operatorname{\mathcal{C}}\ar [d]^{F} \\ \operatorname{\mathcal{D}}' \ar [r] & \operatorname{\mathcal{D}}. } \end{gathered} \end{equation}

where $F$ is an equivalence of $\infty $-categories. Then (4.25) is a categorical pullback square if and only if $F'$ is an equivalence of $\infty $-categories.