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Theorem 9.2.9.15 (Existence of 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 objects $X$ and $Y$ are $\kappa $-compact for some small cardinal $\kappa $.
Then $\operatorname{\mathcal{C}}$ admits a factorization system $( S_{L}, S_{R} )$, where $S_{R}$ is the collection of morphisms of $\operatorname{\mathcal{C}}$ which are right orthogonal to $W$.
Proof.
By virtue of Corollary 9.2.7.21 (and Proposition 
), we can enlarge $W$ to arrange that every morphism $w: A \rightarrow B$ which belongs to $W$ admits a relative codiagonal $\gamma _{A/B}: B \coprod _{A} B \rightarrow B$ which also belongs to $W$. Applying Theorem 9.2.6.7, we conclude that $\operatorname{\mathcal{C}}$ admits a weak factorization system $( S_{L}, S_{R} )$, where $S_{L}$ is the weakly saturated class of morphisms generated by $W$ and $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W$. Using Corollary 9.2.7.21, we see that a morphism $g: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ belongs to $S_{R}$ if and only if it is right orthogonal to $W$. If this condition is satisfied, then Corollary 9.2.7.20 guarantees that $g$ is right orthogonal to $W$. Allowing $g$ to vary, we conclude that $(S_{L}, S_{R} )$ is a factorization system.
$\square$