Exercise 8.5.7.8. Let $(X,x)$ be a pointed Kan complex and let $e: X \rightarrow X$ be a morphism from $X$ to itself. Show that:
If $X$ is connected, then $e$ can be lifted to a morphism $\widetilde{e}: (X,x) \rightarrow (X,x)$ in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$.
If $X$ is simply connected, then $e$ is homotopy idempotent (in the $\infty $-category $\operatorname{\mathcal{S}}$) if and only if $\widetilde{e}$ is homotopy idempotent (in the $\infty $-category $\operatorname{\mathcal{S}}_{\ast }$).
In particular, if $X$ is simply connected, then every homotopy idempotent $e: X \rightarrow X$ is (split) idempotent (Corollary 8.5.7.7).