Kerodon

Example 6.2.3.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}$ be a full subcategory, and let $W$ be the collection of all $\operatorname{\mathcal{C}}'$-local equivalences in $\operatorname{\mathcal{C}}$. Then $W$ automatically satisfies conditions $(1)$ and $(2)$ of Definition 6.2.3.9 (Remarks 6.2.2.4 and 6.2.2.5). If the full subcategory $\operatorname{\mathcal{C}}'$ is reflective, then $W$ is localizing. Similarly, if $\operatorname{\mathcal{C}}'$ is a coreflective subcategory of $\operatorname{\mathcal{C}}$, then the collection of $\operatorname{\mathcal{C}}'$-colocal equivalences is colocalizing.