Proposition 9.2.3.6. Let $\kappa $ be a regular cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-sequentially cocomplete (in the sense of Definition 9.2.3.1).
The $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa , \kappa ^{+} )$-cocomplete (in the sense of Definition 9.2.1.3), where $\kappa ^{+}$ is the successor of $\kappa $.
Proof. The implication $(2) \Rightarrow (1)$ is immediate, since the $\infty $-category $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } )$ is $\kappa ^{+}$-small and $\kappa $-filtered (Example 9.1.1.8). The converse follows from Corollary 7.2.2.12, since every $\kappa ^{+}$-small $\kappa $-filtered $\infty $-category $\operatorname{\mathcal{K}}$ admits a right cofinal functor $\operatorname{N}_{\bullet }( \mathrm{Ord}_{< \kappa } ) \rightarrow \operatorname{\mathcal{K}}$ (Corollary 9.1.8.9). $\square$