Proposition 9.2.9.13 (Lifting Factorization Systems). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of $\infty $-categories and let $(S_{L}, S_{R} )$ be a weak factorization system on $\operatorname{\mathcal{C}}$. Let $\widetilde{S}_{L}$ denote the collection of all $U$-cocartesian morphisms $\widetilde{f}$ in $\operatorname{\mathcal{E}}$ satisfying $U( \widetilde{f} ) \in S_{L}$, and let $\widetilde{S}_{R}$ be the collection of all morphisms $\widetilde{g}$ in $\operatorname{\mathcal{E}}$ satisfying $U( \widetilde{g}) \in S_{R}$. Then the pair $( \widetilde{S}_{L}, \widetilde{S}_{R} )$ is a weak factorization system on $\operatorname{\mathcal{E}}$. If $(S_{L}, S_{R} )$ is a factorization system on $\operatorname{\mathcal{C}}$, then $( \widetilde{S}_{L}, \widetilde{S}_{R} )$ is a factorization system on $\operatorname{\mathcal{E}}$.
Proof. By assumption, the collection $S_{L}$ is weakly left orthogonal to $S_{R}$. Applying Corollary 9.2.7.23, we see that $\widetilde{S}_{L}$ is weakly left orthogonal to $\widetilde{S}_{R}$ (and left orthogonal if the pair $(S_{L}, S_{R} )$ is a factorization system on $\operatorname{\mathcal{C}}$). Since $S_{L}$ and $S_{R}$ are closed under isomorphism, the collections $\widetilde{S}_{L}$ and $\widetilde{S}_{R}$ have the same property (see Corollary 8.5.1.13). We will complete the proof by showing that the pair $( \widetilde{S}_{L}, \widetilde{S}_{R} )$ satisfies condition $(1)$ of Definition 9.2.6.1. Let $\widetilde{h}: \widetilde{X} \rightarrow \widetilde{Z}$ be a morphism in the $\infty $-category $\operatorname{\mathcal{E}}$, and let $h: X \rightarrow Z$ denote its image in the $\infty $-category $\operatorname{\mathcal{C}}$. Since $(S_{L}, S_{R} )$ is a weak factorization system, we can choose a $2$-simplex $\sigma :$
of $\operatorname{\mathcal{C}}$, where $f$ belongs to $S_{L}$ and $g$ belongs to $S_{R}$. Our assumption that $U$ is a cocartesian fibration guarantees that we can lift $f$ to a $U$-cocartesian morphism $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\operatorname{\mathcal{E}}$. Since $\widetilde{f}$ is $U$-cocartesian, we can lift $\sigma $ to a $2$-simplex $\widetilde{\sigma }:$
in the $\infty $-category $\operatorname{\mathcal{E}}$. By construction, we have $\widetilde{f} \in \widetilde{S}_{L}$ and $\widetilde{g} \in \widetilde{S}_{R}$. $\square$