Kerodon

Variant 9.2.1.12. Let $\kappa \leq \lambda $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Suppose that $\operatorname{\mathcal{C}}$ is essentially $\mu $-small, where $\mu > \lambda $ is a regular cardinal of exponential cofinality $\geq \lambda $. Then $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}})$ is also essentially $\mu $-small (Proposition 8.7.3.15). Beware that the inequality $\mu > \lambda $ cannot be replaced by the weaker condition $\mu \geq \lambda $. For example, if $\operatorname{\mathcal{C}}$ is an essentially small $\infty $-category, then $\operatorname{Ind}(\operatorname{\mathcal{C}})$ is usually not essentially small.