Proposition 9.4.1.18. Suppose we are given a commutative diagram of $\infty $-categories
9.33
\begin{equation} \begin{gathered}\label{equation:fiberwise-idempotent-completion2} \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{E}}\ar [dr]_{ U } \ar [rr]^-{H} & & \widehat{\operatorname{\mathcal{E}}} \ar [dl]^{ \widehat{U} } \\ & \operatorname{\mathcal{C}}, & } \end{gathered} \end{equation}
where $U$ and $\widehat{U}$ are inner fibrations. Then (9.33) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise idempotent completion of $\operatorname{\mathcal{E}}$ if and only if it satisfies the following pair of conditions:
- $(1)$
For each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\widehat{\operatorname{\mathcal{E}}}_{C}$ is idempotent complete.
- $(2)$
The functor $H$ is fully faithful.