Variant 4.7.8.10. Let $\lambda $ be an uncountable cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\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 locally $\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 slice $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ is locally $\lambda $-small; the analogous assertion for $\operatorname{\mathcal{C}}_{f/}$ follows by a similar argument. Fix a pair of objects $X,Y \in \operatorname{\mathcal{C}}_{/f}$; we wish to show that the morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}_{/f}}( X, Y)$ is essentially $\lambda $-small. Let $\operatorname{\mathcal{C}}'$ be the smallest full subcategory of $\operatorname{\mathcal{C}}$ which contains $f(K)$ together with the images of $X$ and $Y$. Replacing $\operatorname{\mathcal{C}}$ by $\operatorname{\mathcal{C}}'$, we can reduce to the case where $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small. In this case, the $\infty $-category $\operatorname{\mathcal{C}}_{/f}$ is essentially $\lambda $-small (Corollary 4.7.5.17), so the desired result follows from Example 4.7.8.4. $\square$