Let \(\mathcal C\) be an \(\infty\)-category equipped with a factorization system \((\mathcal L,\mathcal R)\), let \(\mathcal C_0 \subseteq \mathcal C\) be a subcategory, and let us write \(\mathcal L_0 \coloneqq \mathcal L\cap \mathcal C_0\) and \(\mathcal R_0 \coloneqq \mathcal R\cap \mathcal C_0\). Then, the pair \((\mathcal L_0,\mathcal R_0)\) forms a factorization system on \(\mathcal C_0\) if and only if the following conditions are satisfied.
For any solid commutative square ([00JD]) in \(\mathcal C_0\) with \(l \in \mathcal L\) and \(r \in \mathcal R\), the unique lift in \(\mathcal C\) (guaranteed by the orthogonality relation \(\mathcal L\bot \mathcal R\)) also lies in \(\mathcal C_0\).
For any morphism \(c \xrightarrow{f} d\) in \(\mathcal C_0\), both morphisms in the factorization \(c \rightarrow\mathrm{Fact}_{(\mathcal L,\mathcal R)}(f) \rightarrow d\) also lie in \(\mathcal C_0\).
In particular, if \(\mathcal C_0\) is a full subcategory of \(\mathcal C\), then \((\mathcal L_0,\mathcal R_0)\) forms a factorization system if and only if \(\mathrm{Fact}_{(\mathcal L,\mathcal R)}(f)\) lies in \(\mathcal C_0\) for any morphism \(f\) in \(\mathcal C_0\).