Fix an \(\infty\)-category \(\mathcal C\) with a factorization system \((\mathcal L,\mathcal R)\).
For any object \(c \in \mathcal C\), we obtain factorization systems on both \(\mathcal C_{c/}\) and \(\mathcal C_{/c}\) in which both classes are pulled back from \(\mathcal C\) via the respective forgetful functors.
Suppose that \(\mathcal C\) is presentable and that \((\mathcal L,\mathcal R)\) is of small generation. Then, the factorization systems of part ([00K6]) are both of small generation as well. Specifically, if \(S\) denotes a set of morphisms in \(\mathcal C\) that generates \((\mathcal L,\mathcal R)\), then they are respectively generated by the evident (small) spaces of morphisms indexed by \[\bigsqcup_{(a \rightarrow b) \in S} \mathrm{Hom}_\mathcal C(c,a) \qquad \text{and} \qquad \bigsqcup_{(a \rightarrow b) \in S} \mathrm{Hom}_\mathcal C(b,c) ~.\]