Kerodon

Example 9.4.2.9. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-compactly generated. For every simplicial set $K$, the diagram $\infty $-category $\operatorname{Fun}( K, \operatorname{\mathcal{C}})$ is $(\kappa ,\lambda )$-cocomplete and the diagonal map $\delta : \operatorname{\mathcal{C}}\rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})$ is $(\kappa ,\lambda )$-finitary (see Proposition 7.1.8.2). If $K$ is $\kappa $-small, then the functor $\delta $ is $(\kappa ,\lambda )$-compact (see Proposition 9.2.8.9).