Corollary 4.7.5.17. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\lambda $-small, and let $K$ be a simplicial set. Suppose that $K$ is $\kappa $-small, where $\kappa = \mathrm{ecf}(\lambda )$ is the exponential cofinality of $\lambda $. Then, for any diagram $f: K \rightarrow \operatorname{\mathcal{C}}$, the $\infty $-categories $\operatorname{\mathcal{C}}_{f/}$ and $\operatorname{\mathcal{C}}_{/f}$ are essentially $\lambda $-small. Moreover, if $\kappa $ is uncountable, then it suffices to assume that $K$ is essentially $\kappa $-small.
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Proof. We will show that the $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ is essentially $\lambda $-small; the corresponding assertion for $\operatorname{\mathcal{C}}_{f/}$ follows by a similar argument. Theorem 4.6.4.17 supplies an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_{/f} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ f\} $. By virtue of Proposition 4.7.5.14, it will suffice to show that $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is essentially $\lambda $-small, which follows from Remark 4.7.5.10. $\square$