Proposition 9.2.3.5. Let $(Q, \leq )$ be a nonempty linearly ordered set with no largest element, so that the cofinality $\kappa = \mathrm{cf}(Q)$ is a regular cardinal (Example 4.7.3.15). Then an $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-sequentially cocomplete (in the sense of Definition 9.2.3.1) if and only if it admits $\operatorname{N}_{\bullet }(Q)$-indexed colimits. If this condition is satisfied, then a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\kappa $-sequentially cocontinuous if and only if it preserves $\operatorname{N}_{\bullet }(Q)$-indexed colimits.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Recall that the cofinality $\kappa = \mathrm{cf}(Q)$ is characterized by the requirement that there exists a cofinal map of linearly ordered sets $f: \mathrm{Ord}_{< \kappa } \rightarrow Q$. Then, for each $q \in Q$, there exists an ordinal $\alpha < \kappa $ satisfying $q \leq f( \alpha )$. Let $g(q)$ be the smallest such ordinal, so that the construction $q \mapsto g(q)$ determines a cofinal map of linearly ordered sets $g: Q \rightarrow \mathrm{Ord}_{< \kappa }$. Invoking Corollary 7.2.3.4, we deduce that the functors
\[ \operatorname{N}_{\bullet }(f): \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } ) \rightarrow \operatorname{N}_{\bullet }(Q) \quad \quad \operatorname{N}_{\bullet }(g): \operatorname{N}_{\bullet }(Q) \rightarrow \operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } ) \]
are right cofinal. The desired results now follow from Corollaries 7.2.2.12 and 7.2.2.8. $\square$