Corollary 8.5.3.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $F: \operatorname{N}_{\bullet }( \operatorname{Idem}) \rightarrow \operatorname{\mathcal{C}}$ be an idempotent in $\operatorname{\mathcal{C}}$. If $F$ admits a limit (or colimit) in $\operatorname{\mathcal{C}}$, then it is preserved by any functor of $\infty $-categories $G: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Suppose that $F$ admits a limit in $\operatorname{\mathcal{C}}$. Then $F$ splits (Corollary 8.5.3.11): that is, it extends to a diagram $\overline{F}: \operatorname{N}_{\bullet }( \operatorname{Ret}) \rightarrow \operatorname{\mathcal{C}}$. Let $T: \operatorname{Idem}^{\triangleleft } \rightarrow \operatorname{Ret}$ be as in Remark 8.5.3.9, so that $(\overline{F} \circ \operatorname{N}_{\bullet }(T)): \operatorname{N}_{\bullet }( \operatorname{Idem})^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. We wish to show that the functor $(G \circ \overline{F} \circ \operatorname{N}_{\bullet }(T)): \operatorname{N}_{\bullet }( \operatorname{Idem})^{\triangleleft } \rightarrow \operatorname{\mathcal{D}}$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{D}}$. By virtue of Proposition 8.5.3.8, this is automatic (Remark 8.5.3.9). $\square$