Remark 4.1.2.18. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the full simplicial subset of $\operatorname{\mathcal{C}}$ spanned by a set of vertices $S$ of $\operatorname{\mathcal{C}}$. Then a morphism of simplicial sets $f: K \rightarrow \operatorname{\mathcal{C}}$ factors through the simplicial subset $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ if and only if, for every vertex $x \in K$, the image $f(x) \in \operatorname{\mathcal{C}}$ belongs to $S$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$