Proposition 8.5.4.23. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, and let $\operatorname{\mathcal{C}}_{01}$ denote the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ (see Definition 4.6.4.1). Then an idempotent $E: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_{01}$ splits if and only if its images in $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ split. In particular, if $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$ are idempotent complete, then $\operatorname{\mathcal{C}}_{01}$ is idempotent complete.
Proof. Let us identify $E$ with a triple $(E_0, E_1, u)$, where $E_0: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_0$ and $E_1: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}_1$ are idempotents in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, and $u: (F_0 \circ E_0) \rightarrow (F_1 \circ E_1)$ is a natural transformation. Suppose that the idempotents $E_0$ and $E_1$ split; we wish to show that $E$ splits (the reverse implication is trivial). Choose functors $\overline{E}_0: \operatorname{N}_{\bullet }(\operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_0$ and $\overline{E}_1: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_1$ extending $E_0$ and $E_1$, respectively. It follows from Corollary 8.5.3.10 that $u$ admits an (essentially unique) extension to a natural transformation $\overline{u}: (F_0 \circ \overline{E}_0) \rightarrow (F_1 \circ \overline{E}_1)$. The triple $(\overline{E}_0, \overline{E}_1, \overline{u} )$ can then be regarded as a diagram $\operatorname{N}_{\bullet }(\operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}_{01}$ extending $E$. $\square$