Kerodon

Remark 9.6.9.17. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $\overline{W}$ be a saturated collection of morphisms of $\operatorname{\mathcal{C}}$. Then $\overline{W}$ is accessibly saturated (in the sense of Remark 9.6.9.6) if and only if it is accessibly weakly saturated (in the sense of Variant 9.6.9.15). In other words, if there exists a small subset $W \subseteq \overline{W}$ which generates $\overline{W}$ as a saturated collection of morphisms, then there is another small subset $W^{+} \subseteq \overline{W}$ which generates $\overline{W}$ as a weakly saturated class of morphisms. For example, we can take $W^{+}$ to be the collection of morphisms appearing in the proof of Theorem 9.6.9.8.