Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 9.2.7.20. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $g: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$, and let $S$ be the collection of morphisms of $\operatorname{\mathcal{C}}$ which are left orthogonal to $g$. Then $S$ is weakly saturated.

Proof. Combining Corollaries 9.2.7.14, 9.2.7.15, and 9.2.7.19 with Proposition 9.2.2.10 (and Remark 9.2.2.11), we see that $S$ is closed under transfinite composition. Since $S$ is also closed under retracts (Corollary 9.2.7.17) and pushouts (Corollary 9.2.7.18), it is weakly saturated. $\square$