Kerodon

Remark 1.5.4.14. Let $\operatorname{\mathcal{C}}$ be a category and let $S_0$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. Then there exists a smallest collection of morphisms $S$ of $\operatorname{\mathcal{C}}$ such that $S_0 \subseteq S$ and $S$ is weakly saturated (for example, we can take $S$ to be the intersection of all the weakly saturated collections of morphisms containing $S_0$). We will refer to $S$ as the weakly saturated collection of morphisms generated by $S_0$. It follows from Proposition 1.5.4.13 that if $S_0$ is weakly left orthogonal to some collection of morphisms $T$, then $S$ has the same property.