Warning 8.5.6.18. The converse of Proposition 8.5.6.17 is false. For example, every isomorphism $e: X \rightarrow X$ is weakly split, but is split idempotent only if $e$ is homotopic to the identity morphism $\operatorname{id}_{X}$ (see Example 8.5.2.2).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$