Warning 5.4.6.7. The analogue of Proposition 5.4.6.6 for the two-out-of-three property is false in general. For example, if $\operatorname{\mathcal{C}}$ is the nerve of a category, then the collection of identity morphisms of $\operatorname{\mathcal{C}}$ has the two-out-of-three property, but usually does not contain all the isomorphisms of $\operatorname{\mathcal{C}}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$