Kerodon

Example 9.4.2.6. Let $\kappa \leq \lambda $ be regular cardinals and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an equivalence of $\infty $-categories. Then $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated if and only if $\operatorname{\mathcal{D}}$ is $(\kappa ,\lambda )$-compactly generated. If these conditions are satisfied, then the functor $F$ is automatically $(\kappa ,\lambda )$-compact. In particular, if $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, then the identity functor $\operatorname{id}_{\operatorname{\mathcal{C}}}$ is $(\kappa ,\lambda )$-compact.