Remark 8.5.6.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an split idempotent endomorphism in $\operatorname{\mathcal{C}}$, so that the diagram
\[ \cdots \rightarrow X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \xrightarrow {e} X \rightarrow \cdots \]
admits both a limit and colimit in $\operatorname{\mathcal{C}}$. The limit and colimit of this diagram are automatically preserved by any functor of $\infty $-categories $\operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$. This follows by combining Corollary 8.5.3.12 with Proposition 8.5.4.16.