Corollary 8.5.4.24. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left fibration of $\infty $-categories. If $\operatorname{\mathcal{C}}$ is idempotent complete, then $\operatorname{\mathcal{E}}$ is idempotent complete.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Choose an uncountable regular cardinal $\kappa $ such that $U$ is essentially $\kappa $-small, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa }$ be a covariant transport representation for $U$. Then the $\infty $-category $\operatorname{\mathcal{E}}$ is equivalent to the oriented fiber $\{ \Delta ^0 \} \operatorname{\vec{\times }}_{\operatorname{\mathcal{S}}} \operatorname{\mathcal{C}}$, which is idempotent complete by virtue of Proposition 8.5.4.23. $\square$