Warning 9.2.1.23. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which is both localizing and colocalizing. In this case, Proposition 9.2.1.18 and Variant 9.2.1.22 provide two different concrete realizations of the localization $\operatorname{\mathcal{C}}[W^{-1}]$, given by the full subcategories $\operatorname{\mathcal{C}}' \subseteq \operatorname{\mathcal{C}}\supseteq \operatorname{\mathcal{C}}''$ spanned by the $W$-local and $W$-colocal objects of $\operatorname{\mathcal{C}}$, respectively. Note that $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}''$ are necessarily equivalent as abstract $\infty $-categories. More precisely, if $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$ is a functor which exhibits $\operatorname{\mathcal{C}}[W^{-1}]$ as the localization of $\operatorname{\mathcal{C}}$ with respect to $W$, then the restrictions
are equivalences of $\infty $-categories. Beware that $\operatorname{\mathcal{C}}'$ and $\operatorname{\mathcal{C}}''$ usually do not coincide when regarded as subcategories of $\operatorname{\mathcal{C}}$. See Warning 6.3.3.12.