Warning 3.5.4.15. The terminology of Definition 3.5.4.14 is potentially confusing. If $f: X \rightarrow Y$ is a morphism which exhibits $Y$ as an $n$-coskeleton of $X$, then it generally does not exhibit $Y$ as a weak $n$-coskeleton of $X$ (because $f$ need not be surjective on $(n+1)$-simplices).
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$