Corollary 4.7.5.14. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then any full subcategory $\operatorname{\mathcal{C}}_0 \subseteq \operatorname{\mathcal{C}}$ is essentially $\kappa $-small.
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Proof. Let $\operatorname{\mathcal{C}}_1 \subseteq \operatorname{\mathcal{C}}$ be the full subcategory spanned by those objects $X \in \operatorname{\mathcal{C}}$ which are isomorphic to an object of $\operatorname{\mathcal{C}}_0$. Proposition 4.7.5.12 guarantees that $\operatorname{\mathcal{C}}_1$ is essentially $\kappa $-small. Since the inclusion $\operatorname{\mathcal{C}}_0 \hookrightarrow \operatorname{\mathcal{C}}_1$ is an equivalence of $\infty $-categories, it follows that $\operatorname{\mathcal{C}}_0$ is also essentially $\kappa $-small (Remark 4.7.5.2). $\square$