Kerodon

Example 4.7.8.4. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is essentially $\kappa $-small. Then $\operatorname{\mathcal{C}}$ is locally $\kappa $-small: that is, 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. This is a special case of Proposition 4.7.5.14, since $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ can be identified with the oriented fiber product $\{ X\} \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \{ Y\} $.