Proposition 6.3.2.13. Let $\kappa $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then any localization of $\operatorname{\mathcal{C}}$ is also essentially $\kappa $-small.
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Proof. Without loss of generality, we may assume that $\operatorname{\mathcal{C}}$ is $\kappa $-small. If $\operatorname{\mathcal{C}}[W^{-1}]$ is a localization of $\operatorname{\mathcal{C}}$ with respect to some collection of morphisms $W$, then there is a categorical equivalence of simplicial sets $\operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}[W^{-1}]$, where $\operatorname{\mathcal{D}}$ is the localization of Corollary 6.3.2.9. Since $\operatorname{\mathcal{D}}$ is $\kappa $-small (Remark 6.3.2.10), it follows that $\operatorname{\mathcal{C}}[W^{-1}]$ is essentially $\kappa $-small (see Proposition 4.7.5.5). $\square$