Kerodon

Remark 9.3.5.8. Let $\operatorname{\mathcal{C}}$ be an $\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 weakly right orthogonal to every morphism of $W$, and let $\overline{W}$ denote the collection of all morphisms which are weakly left orthogonal to every morphism of $W_{\perp }$. Then $\overline{W}$ is always a weakly saturated collection of morphisms which contains $W$ (Corollary 9.3.4.17). If the hypotheses of Theorem 9.3.5.7 are satisfied, then $\overline{W}$ is the weakly saturated collection generated by $W$, in the sense of Remark 9.3.2.26.