Remark 8.5.7.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $e: X \rightarrow X$ be an endomorphism in $\operatorname{\mathcal{C}}$. Then $e$ is homotopy idempotent if and only if there exists a $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ whose boundary is indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & X \ar [dr]^{e} & \\ X \ar [ur]^{e} \ar [rr]^{e} & & X. } \]