$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$. The following conditions are equivalent:


The idempotent $F$ is split: that is, it can be extended to a functor $\operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$.


The diagram $F$ admits a limit in $\operatorname{\mathcal{C}}$.


The diagram $F$ admits a colimit in $\operatorname{\mathcal{C}}$.

Proof. The implications $(1) \Rightarrow (2)$ and $(1) \Rightarrow (3)$ follow from Remark, and the reverse implications follow from Corollary $\square$