Kerodon

Proof. For each morphism $w: A \rightarrow B$ of $\operatorname{\mathcal{C}}$, choose a relative codiagonal $\gamma _{w}: B \coprod _{A} B \rightarrow B$ (see Variant 7.6.2.23). Let $W^{+}$ be the smallest collection of morphisms which contains $W$ and is closed under the operation $w \mapsto \gamma _{w}$. It follows from Theorem 9.6.5.12 that $\operatorname{\mathcal{C}}$ admits a weak factorization system $(S_{L}, S_{R})$, where $S_{L}$ is the weakly saturated collection of morphisms generated by $W^{+}$ and $S_{R}$ is the collection of morphisms which are weakly right orthogonal to $W^{+}$. Using Corollaries 9.6.6.15 and Corollary 9.6.6.26, we see that a morphism of $\operatorname{\mathcal{C}}$ belongs to $S_{R}$ if and only if it is right orthogonal to $W$. Applying Corollary 9.6.6.15 again, we see that $S_{R}$ is closed under the formation of relative diagonals, so that $(S_ L, S_ R)$ is a factorization system on $\operatorname{\mathcal{C}}$ (Corollary 9.6.8.14). In particular, $S_{L}$ is a saturated collection of morphisms of $\operatorname{\mathcal{C}}$ (Example 9.6.9.4). To complete the proof, it will suffice to show that if $S$ is another saturated collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $W$, then it contains $S_{L}$. This is clear, since $S$ is weakly saturated (Remark 9.6.9.7) and contains $W^{+}$ (Remark 9.6.9.5). $\square$