Theorem 9.6.5.12 (Existence of Weak Factorization Systems). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $W$ be a small collection of morphisms of $\operatorname{\mathcal{C}}$. Then $\operatorname{\mathcal{C}}$ admits a weak factorization system $(S_ L, S_ R)$, where $S_{L}$ is the weakly saturated collection of morphisms generated by $W$ (Remark 9.6.5.11) and $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W$.
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Proof of Theorem 9.6.5.12. The collection $S_{L}$ is closed under retracts by construction, and $S_{R}$ is closed under retracts by virtue of Proposition 9.6.4.14. Example 9.6.5.9 guarantees that $S_{L}$ is weakly left orthogonal to $S_{R}$. It will therefore suffice to show that every morphism $h: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ factors as a composition $X \xrightarrow {f} Y \xrightarrow {g} Z$, where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$.
Let $\widetilde{\operatorname{\mathcal{C}}}$ denote the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Z}$, and let $\pi : \widetilde{\operatorname{\mathcal{C}}} \rightarrow \operatorname{\mathcal{C}}$ denote the projection map. Our assumption that $\operatorname{\mathcal{C}}$ is presentable guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ is also presentable (Proposition 9.5.4.1). Let $\widetilde{W}$ denote a set of representatives for the collection of isomorphism classes of morphisms $\widetilde{w}$ of $\operatorname{\mathcal{C}}_{/Z}$ satisfying $\pi ( \widetilde{w} ) \in W$. Since $W$ is small and $\operatorname{\mathcal{C}}$ is locally small, the set $\widetilde{W}$ is also small. Applying Theorem 9.6.3.3, we deduce that there is a morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\widetilde{\operatorname{\mathcal{C}}}$ which is a transfinite pushout of morphisms of $\widetilde{W}$, where $\widetilde{Y}$ is weakly $\widetilde{W}$-local. Set $f = \pi (\widetilde{f})$, so that $\widetilde{f}$ can be identified with a diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]_{g} & \\ X \ar [ur]^{f} \ar [rr]^{h} & & Z } \]
in the $\infty $-category $\operatorname{\mathcal{C}}$. Since the functor $\pi $ preserves small colimits (Corollary 7.1.4.27), the morphism $f$ is a transfinite pushout of morphisms belonging to $W$, and therefore belongs to $S_{L}$. Our assumption that $\widetilde{Y}$ is weakly $\widetilde{W}$-local guarantees that $g$ belongs to $S_{R}$ (Remark 9.6.4.12). $\square$