Remark 3.5.1.23. In the situation of Proposition 3.5.1.22, it is not necessary to verify that $X_{y}$ is $n$-connective for every vertex $y \in Y$; it is enough to check this condition at one vertex in each connected component of $Y$ (see Remark 3.3.7.3). In particular, if $Y$ is connected, then it is enough to check that $X_{y}$ is $n$-connective for any vertex $y \in Y$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$