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Proposition 7.6.6.24. Let $\lambda $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which arises as the limit of a diagram

\[ A \rightarrow \operatorname{\mathcal{QC}}\quad \quad (\alpha \in A) \mapsto \operatorname{\mathcal{C}}_{\alpha }. \]

Assume that:

$(a)$

For each vertex $\alpha \in A$, the $\infty $-category $\operatorname{\mathcal{C}}_{\alpha }$ is $\mathbb {K}$-complete.

$(b)$

For each edge $e: \alpha \rightarrow \alpha '$ of $A$, the transition map $\operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{C}}_{\alpha '}$ is a $\mathbb {K}$-continuous functor.

Then the $\infty $-category $\operatorname{\mathcal{C}}$ is $\mathbb {K}$-complete. Moreover, a functor of $\infty $-categories $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is $\mathbb {K}$-continuous if and only if, for each vertex $\alpha \in A$, the composite functor $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}_{\alpha }$ is $\mathbb {K}$-continuous.