Theorem 9.1.6.7 (Existence of Weak Factorization Systems). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Assume that:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is locally small and admits small colimits.
- $(2)$
The collection $W$ is small.
- $(3)$
For every morphism $w: X \rightarrow Y$ in $W$, the object $X$ is $\kappa $-compact for some small cardinal $\kappa $.
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 9.1.3.23) and $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W$.
Proof of Theorem 9.1.6.7.
The collection $S_{L}$ is closed under retracts by construction, and $S_{R}$ is closed under retracts by virtue of Proposition 9.1.5.14. Corollary 9.1.5.17 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 locally small guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ is also locally small (Example 4.7.8.11), and our assumption that $\operatorname{\mathcal{C}}$ admits small colimits guarantees that $\widetilde{\operatorname{\mathcal{C}}}$ admits small colimits (Corollary 7.1.3.20). 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. For every morphism $\widetilde{w}: \widetilde{A} \rightarrow \widetilde{B}$ which belongs to $\widetilde{W}$, the image $A = \pi ( \widetilde{A} )$ is a $\kappa $-compact object of $\widetilde{\operatorname{\mathcal{C}}}$ for some small cardinal $\kappa $, so that $\widetilde{A}$ is a $\kappa $-compact object of $\widetilde{\operatorname{\mathcal{C}}}$ (Remark 
). Let us identify the morphism $h$ with an object $\widetilde{X} \in \widetilde{\operatorname{\mathcal{C}}}$ satisfying $\pi ( \widetilde{X} ) = X$. Applying Theorem 9.1.4.3, 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 $\widetilde{W}$-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 7.1.3.20), the morphism $f$ is a transfinite pushout of morphisms belonging to $W$, and therefore belongs to $S_{L}$. Our assumption that $\widetilde{Y}$ is $\widetilde{W}$-local guarantees that $g$ belongs to $S_{R}$ (Remark 9.1.5.12).
$\square$