Example 9.4.1.17 (Fiberwise Idempotent Completion). Suppose we are given a commutative diagram of simplicial sets
9.32
\begin{equation} \begin{gathered}\label{equation:fiberwise-idempotent-completion} \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. We say that (9.32) exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a fiberwise idempotent completion of $\operatorname{\mathcal{E}}$ if it exhibits $\widehat{\operatorname{\mathcal{E}}}$ as a $\{ \operatorname{N}_{\bullet }( \operatorname{Idem}) \} $-cocompletion of $\operatorname{\mathcal{E}}$, in the sense of Definition 9.4.1.12; here $\operatorname{Idem}$ is the category introduced in Construction 8.5.2.7.