Kerodon

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

Proposition 9.2.9.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(S_{L}, S_{R})$ be a factorization system on $\operatorname{\mathcal{C}}$, and let $f$ be a morphism of $\operatorname{\mathcal{C}}$. The following conditions are equivalent:

$(1)$

The morphism $f$ belongs to $S_{L}$.

$(2)$

The morphism $f$ is left orthogonal to $S_{R}$.

$(3)$

The morphism $f$ is weakly left orthogonal to $S_{R}$.

Proof. The implication $(1) \Rightarrow (2)$ is immediate from the definition, the implication $(2) \Rightarrow (3)$ follows from Remark 9.2.7.7, and the implication $(3) \Rightarrow (1)$ follows from Proposition 9.2.6.5 (together with Corollary 9.2.9.9). $\square$