Definition 4.7.8.1. Let $\kappa $ be an uncountable cardinal. We say that an $\infty $-category $\operatorname{\mathcal{C}}$ is locally $\kappa $-small if, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is essentially $\kappa $-small.
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