Remark 8.5.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then an endomorphism $e: X \rightarrow X$ is split idempotent if and only there exists a retraction diagram
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{r} & \\ Y \ar [ur]^{i} \ar [rr]^{ \operatorname{id}_{Y} } & & Y. } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$, where $e$ factors as a composition $X \xrightarrow {r} Y \xrightarrow {i} X$.