Fix an \(\infty\)-category \(\mathcal B\) and a functor \(\mathcal B^\mathrm{op}\xrightarrow{F} \widehat{\mathrm{Cat}}_\infty^{\text{f.s.}, \mathcal R}\) (recall Definition B.1.5). For each \(b \in \mathcal B\), let us write \((\mathcal L_b,\mathcal R_b)\) for the given factorization system on \(F(b)\). Moreover, let us write \(\mathcal E\xrightarrow{p} \mathcal B\) for the Cartesian fibration associated to \(F\).
The \(\infty\)-category \(\mathcal E\) admits a factorization system \((\mathcal L,\mathcal R)\), described as follows.
A morphism \(e \xrightarrow{\alpha} f\) lies in \(\mathcal L\) if and only if the morphism \(p(e) \xrightarrow{p(\alpha)} p(f)\) in \(\mathcal B\) is an equivalence and moreover the morphism \(e \rightarrow p(\alpha)^*(f)\) in \(\mathcal E_{p(e)} \simeq F(p(e))\) lies in \(\mathcal L_{p(e)}\).
A morphism \(e \xrightarrow{\alpha} f\) lies in \(\mathcal R\) if and only if the morphism \(e \rightarrow p(\alpha)^*(f)\) lies in \(\mathcal R_{p(e)}\).
Suppose that \(\mathcal B\) is small and that \(F\) factors through the subcategory \(\Pr^{R,\text{f.s.},\mathcal R} \subset \widehat{\mathrm{Cat}}_\infty^{\text{f.s.},\mathcal R}\) (recall Definition B.1.15). Then, the factorization system of part ([00KP]) is also of small generation. More specifically, if for each \(b \in \mathcal B\) the set \(S_b\) generates the class \(\mathcal L_b\), then the set \(S \coloneqq \bigsqcup_{b \in \mathcal B} S_b\) generates the class \(\mathcal L\) (considering each \(S_b\) as defining a set of morphisms in the fiber \(\mathcal E_b \simeq F(b)\)).