Variant 9.6.9.15 (Accessible Weak Factorization Systems). Corollary 9.6.9.14 has a counterpart for weak factorization systems. Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category. Let us say that a weak factorization system $(S_ L, S_ R)$ on $\operatorname{\mathcal{C}}$ is accessible if there exists a small collection of morphisms $W$ of $\operatorname{\mathcal{C}}$ such that $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W$, and let us say that a collection of morphisms $\overline{W}$ of $\operatorname{\mathcal{C}}$ is accessibly weakly saturated if it is the weakly saturated collection of morphisms generated by some small subset $W \subseteq \overline{W}$. It follows from Theorem 9.6.5.12 that the construction $(S_ L, S_ R) \mapsto S_{L}$ induces a bijection
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Accessible weak factorization systems $(S_ L,S_ R)$ on $\operatorname{\mathcal{C}}$} \} \ar [d]^{\sim } \\ \{ \textnormal{Accessibly weakly saturated collections of morphisms $\overline{W}$ of $\operatorname{\mathcal{C}}$} \} . } \]