Proposition 9.6.9.12. Let $\operatorname{\mathcal{C}}$ be an accessible $\infty $-category and let $(S_ L, S_ R)$ be a factorization system on $\operatorname{\mathcal{C}}$. Then $(S_ L, S_ R)$ is an accessible factorization system if and only if it satisfies the following condition:
There exists a small collection $W$ of morphisms of $\operatorname{\mathcal{C}}$ such that a morphism $g$ of $\operatorname{\mathcal{C}}$ belongs to $S_{R}$ if and only if it is right orthogonal to $W$.
Proof. Let $\mathcal{L}, \mathcal{R} \subseteq \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ be as in Proposition 9.6.9.10. If the factorization system $(S_ L, S_ R)$ is accessible, then $\mathcal{L}$ is an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. Let $W \subseteq S_{L}$ be the collection of morphisms which are $\kappa $-compact when viewed as objects of $\mathcal{L}$. Then every morphism belonging to $S_{L}$ can be realized as a small $\kappa $-filtered colimit of morphisms belonging to $W$. Applying Proposition 9.6.6.12. we see that a morphism $g: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ is right orthogonal to $W$ if and only if it is right orthogonal to $S_{L}$: that is, if and only if it belongs to $S_{R}$ (Proposition 9.6.8.13).
We now prove the converse. For every collection $W$ of morphisms of $\operatorname{\mathcal{C}}$, let $\mathcal{R}_{W}$ denote the full subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$ spanned by those morphisms which are right orthogonal to $W$. We will show that if $W$ is small, then $\mathcal{R}_{W}$ is an accessibly embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. In particular, if $W$ satisfies condition $(\ast )$, then $\mathcal{R} = \mathcal{R}_{W}$ is an accessible embedded subcategory of $\operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$, so that $(S_ L, S_ R)$ is an accessible factorization system.
Using Corollary 9.4.8.13, we can reduce to the case where $W = \{ w\} $ consists of a single morphism $w: C \rightarrow D$ of $\operatorname{\mathcal{C}}$. Let $T: \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{S}}) \rightarrow \operatorname{Fun}( \Delta ^1 \times \Delta ^1, \operatorname{\mathcal{S}})$ denote the functor which carries each morphism $g: X \rightarrow Y$ to the diagram
Note that the functor $T$ is accessible (see Example 9.4.7.16 and Remark 9.4.8.2), and that a morphism $g$ of $\operatorname{\mathcal{C}}$ is right orthogonal to $W$ if and only if $T(g)$ is a pullback diagram in $\operatorname{\mathcal{C}}$. The desired result now follows from Corollary 9.4.8.12. $\square$