Corollary 8.5.6.18. Suppose we are given a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [r]^-{F} \ar [d]^{U} & \operatorname{\mathcal{E}}' \ar [d]^{U'} \\ \operatorname{\mathcal{C}}\ar [r]^-{ \overline{F} } & \operatorname{\mathcal{C}}', } \]
where the vertical maps are right fibrations. If $\overline{F}$ exhibits $\operatorname{\mathcal{C}}'$ as an idempotent completion of $\operatorname{\mathcal{C}}$, then $F$ exhibits $\operatorname{\mathcal{E}}'$ as an idempotent completion of $\operatorname{\mathcal{E}}$.