$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Remark Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be the subcategory spanned by the collection of objects $S$ of $\operatorname{\mathcal{C}}$ and a collection of morphisms $T$ of $\operatorname{\mathcal{C}}$. Then a morphism of simplicial sets $f: K \rightarrow \operatorname{\mathcal{C}}$ factors through the subcategory $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ if and only if it carries each vertex of $K$ to an element of $S$ and each edge of $K$ to an element of $T$.