Remark 4.1.2.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a subcategory. Suppose that $\operatorname{\mathcal{C}}$ contains a $2$-simplex $\sigma :$
which witnesses $h$ as a composition of $g$ and $f$ (Definition 1.4.4.1). If $f$ and $g$ belong to the subcategory $\operatorname{\mathcal{C}}'$, then the $2$-simplex $\sigma $ also belongs to the subcategory $\operatorname{\mathcal{C}}'$ (since the inclusion $\operatorname{\mathcal{C}}' \hookrightarrow \operatorname{\mathcal{C}}$ is weakly right orthogonal to the horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$). In particular, if $f$ and $g$ belong to $\operatorname{\mathcal{C}}'$, then $h$ also belongs to $\operatorname{\mathcal{C}}'$.