Proposition 9.5.4.1 (Presentability of Slice and Coslice $\infty $-Categories). Let $\operatorname{\mathcal{C}}$ be a presentable $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a small diagram. Then the slice and coslice $\infty $-categories $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are also presentable.
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Proof. It follows from Corollary 9.4.8.6 that $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are accessible. It will therefore suffice to show that $\operatorname{\mathcal{C}}_{/f}$ and $\operatorname{\mathcal{C}}_{f/}$ are cocomplete, which follows from Corollary 7.1.4.27 and Remark 7.1.3.16, respectively. $\square$