Kerodon

Example 9.2.9.18. Let $\kappa \leq \lambda $ be regular cardinals, let $\operatorname{\mathcal{C}}$ be a $(\kappa ,\lambda )$-cocomplete $\infty $-category. For every simplicial set $K$, we can identify $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ with the $\infty $-category of sections of the $(\kappa ,\lambda )$-cocomplete inner fibration $K \times \operatorname{\mathcal{C}}\rightarrow K$. In this case, Proposition 9.2.9.16 recovers the conclusion of Proposition 9.2.8.9: if $K$ is $\kappa $-small, then any diagram of $(\kappa ,\lambda )$-compact objects of $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact when viewed as an object of $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$.