$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\lambda $ be an infinite cardinal which is not regular. The following conditions are equivalent:


The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda $-small limits.


For every infinite cardinal $\kappa < \lambda $, the $\infty $-category $\operatorname{\mathcal{C}}$ admits $\kappa $-small limits.


The $\infty $-category $\operatorname{\mathcal{C}}$ admits $\lambda ^{+}$-small limits, where $\lambda ^{+}$ denotes the successor of $\lambda $.

Proof of Corollary The equivalence $(1) \Leftrightarrow (2)$ and the implication $(3) \Rightarrow (1)$ follow from Remark We will complete the proof by showing that $(1)$ implies $(3)$. Assume that $\operatorname{\mathcal{C}}$ admits $\lambda $-small limits; we wish to show that it admits $\lambda ^{+}$-small limits. By virtue of Proposition, it will suffice to show that every collection of objects $\{ X_ i \} _{i \in I}$ admits a product in $\operatorname{\mathcal{C}}$, provided that the index set $I$ has cardinality $\leq \lambda $. Our assumption that $\lambda $ is not regular guarantees that we can decompose $I$ as a disjoint union of $\lambda $-small subsets $\{ I_ j \subseteq I \} _{j \in J}$, where the index set $J$ is $\lambda $-small. It follows from $(1)$ that $\operatorname{\mathcal{C}}$ admits $J$-indexed products and also that it admits $I_{j}$-indexed products for each $j \in J$, and therefore admits $I$-indexed products by virtue of Corollary $\square$