Remark 9.6.9.24. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\overline{W}$ be a strongly saturated collection of morphisms of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(a)$
The collection $\overline{W}$ is accessibly strongly saturated: that is, there exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ as a strongly saturated collection of morphisms.
- $(b)$
The collection $\overline{W}$ is accessibly saturated: that is, there exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ as a saturated collection of morphisms.
- $(c)$
The collection $\overline{W}$ is acccessibly weakly saturated: that is, there exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ as a weakly saturated collection of morphisms.
The implications $(c) \Rightarrow (b) \Rightarrow (a)$ are trivial, the implication $(b) \Rightarrow (c)$ is a special case of Remark 9.6.9.17, and the implication $(a) \Rightarrow (b)$ follows from Proposition 9.6.9.23.