Corollary 8.1.2.17. Let $\operatorname{\mathcal{C}}$ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small, then $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is essentially $\kappa $-small.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Choose an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}'$ is a $\kappa $-small simplicial set. It follows from Corollary 8.1.2.16 that the induced map $\operatorname{Tw}(\operatorname{\mathcal{C}}') \rightarrow \operatorname{Tw}(\operatorname{\mathcal{C}})$ is also an equivalence of $\infty $-categories. We conclude by observing that $\operatorname{Tw}(\operatorname{\mathcal{C}}')$ is also a $\kappa $-small simplicial set (Remark 8.1.1.5). $\square$