Corollary 8.5.7.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits sequential limits and colimits, and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ is idempotent if and only if it is homotopy idempotent and the composite map
\[ \varprojlim ( \cdots \rightarrow X \xrightarrow {e} X \rightarrow \cdots ) \rightarrow X \rightarrow \varinjlim ( \cdots \rightarrow X \xrightarrow {e} X \rightarrow \cdots ) \]
is an isomorphism (see Definition 8.5.6.14).