Kerodon

Remark 9.6.9.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category $\infty $-category, let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$, let $W_{\perp }$ denote the collection of morphisms of $\operatorname{\mathcal{C}}$ which are right orthogonal to every morphism of $W$, and let $\overline{W}$ be the collection of all morphisms which are left orthogonal to $W_{\perp }$. It follows from Example 9.6.9.3 that $\overline{W}$ is a saturated collection of morphisms which contains $W$. If $\operatorname{\mathcal{C}}$ is presentable and $W$ is small, then Theorem 9.6.9.8 guarantees that $\overline{W}$ is the smallest saturated collection of morphisms which contains $W$.