Kerodon

Warning 11.4.0.1. Let $\kappa \leq \lambda $ be regular cardinals, where $\lambda $ is uncountable, and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-accessible. If $\lambda $ is not strongly inaccessible, then $\operatorname{\mathcal{C}}$ need not be locally $\lambda $-small. However, the morphism $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ is essentially $\lambda $-small whenever the object $X \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact. This follows immediately from the definition if $Y$ is $(\kappa ,\lambda )$-compact, and follows in general from the observation that the functor $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, \bullet )$ is $(\kappa ,\lambda )$-finitary.