Proposition 9.2.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $(S_{L}, S_{R})$ be a weak factorization system on $\operatorname{\mathcal{C}}$, and let $h: X \rightarrow Z$ be a morphism of $\operatorname{\mathcal{C}}$. Then $h$ belongs to $S_{L}$ if and only if it is weakly left orthogonal to $S_{R}$, and $h$ belongs to $S_{R}$ if and only if it is weakly right orthogonal to $S_{L}$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. We will prove the first assertion; the second follows by a similar argument. Assume that $h$ is weakly left orthogonal to $S_{R}$; we wish to show that $h$ belongs to $S_{L}$ (the reverse implication is immediate from the definition). By virtue of Remark 9.2.6.4, we may assume that $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category. The morphism $h$ admits a factorization $X \xrightarrow {f} Y \xrightarrow {g} Z$, where $f \in S_{L}$ and $g \in S_{R}$. Since $h$ is weakly left orthogonal to $g$, the lifting problem
\[ \xymatrix@R =50pt@C=50pt{ X \ar [d]^{h} \ar [r]^-{f} & Y \ar [d]^{g} \\ Z \ar@ {-->}[ur]^-{i} \ar [r]^-{\operatorname{id}} & Z } \]
admits a solution. We then have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [r]^-{ \operatorname{id}} \ar [d]^{h} & X \ar [d]^{f} \ar [r]^-{ \operatorname{id}} & X \ar [d]^{h} \\ Z \ar [r]^-{i} & Y \ar [r]^-{ g} & Z } \]
which exhibits $h$ as a retract of $f$, so that $h$ also belongs to $S_{L}$. $\square$