Kerodon

Remark 7.6.6.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\lambda $ be an infinite cardinal. If $\operatorname{\mathcal{C}}$ is $\lambda $-complete, then it is also $\kappa $-complete for each infinite cardinal $\kappa < \lambda $. Similarly, if a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\lambda $-small limits, then it also preserves $\kappa $-small limits for each $\kappa < \lambda $. In both cases, the converse holds if $\lambda $ is an uncountable limit cardinal (since, in that case, every $\lambda $-small simplicial set $K$ is $\kappa $-small for some $\kappa < \lambda $).