Proposition 8.5.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an idempotent endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ splits if and only if the diagram $T_{e}: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{\mathcal{C}}$ admits a limit.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Since $e$ is idempotent, it can be extended to a functor $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$. Then $T_{e} = F \circ Q$, where $Q: \operatorname{Spine}[\operatorname{\mathbf{Z}}] \rightarrow \operatorname{N}_{\bullet }( \operatorname{Idem})$ is the left cofinal morphism of Proposition 8.5.4.16. Using Corollary 7.2.2.10, we see that $T_{e}$ has a limit in $\operatorname{\mathcal{C}}$ if and only if $F$ has a limit in $\operatorname{\mathcal{C}}$. The desired result now follows from the criterion of Corollary 8.5.3.11 $\square$