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

Corollary 4.7.5.16. Let $\kappa $ be an uncountable cardinal and suppose we are given a categorical pullback diagram of $\infty $-categories

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\begin{equation} \begin{gathered}\label{equation:categorical-pullback-essentially-small} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_{01} \ar [r] \ar [d] & \operatorname{\mathcal{C}}_0 \ar [d] \\ \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{\mathcal{C}}. } \end{gathered} \end{equation}

If $\operatorname{\mathcal{C}}_0$, $\operatorname{\mathcal{C}}$, and $\operatorname{\mathcal{C}}_1$ are essentially $\kappa $-small, then $\operatorname{\mathcal{C}}_{01}$ is essentially $\kappa $-small.