Kerodon

Corollary 6.3.1.22. Let $\operatorname{\mathcal{C}}$ be a simplicial set, let $W$ and $W'$ be collections of edges of $\operatorname{\mathcal{C}}$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a morphism of simplicial sets which exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W$. Suppose that, for every edge $w \in W'$, the image $F(w)$ is a degenerate edge of $\operatorname{\mathcal{D}}$. Then $F$ also exhibits $\operatorname{\mathcal{D}}$ as a localization of $\operatorname{\mathcal{C}}$ with respect to $W \cup W'$.