Remark 9.6.9.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. If $S$ is saturated (in the sense of Definition 9.6.9.2), then it is weakly saturated (in the sense of Definition 9.6.5.6). That is:
For any pushout diagram
\[ \xymatrix@C =50pt@R=50pt{ C \ar [d]^{w} \ar [r] & C' \ar [d]^{w'} \\ D \ar [r] & D' } \]in the $\infty $-category $\operatorname{\mathcal{C}}$, if $w$ belongs to $S$, then $w'$ also belongs to $S$. This follows from the observation that $w'$ is a pushout of $w$ and $\operatorname{id}_{C'}$ along $\operatorname{id}_{C}$.
The collection $S$ is closed under the formation of retracts (in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$), since every retract can be written as a colimit indexed by the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{Idem})$ (Remark 8.5.3.9).
The collection $S$ is closed under small transfinite composition: see Proposition 9.6.1.10.