Limits of categories with limits

Corollary 23.5.6. Let \(I\) and \(K\) be small \(\infty \)-categories and let \(C_{\bullet }\colon K \to \Cat _{\infty }\) be a \(K\)-indexed diagram of small \(\infty \)-categories. Assume that \(C_k\) admits \(I\)-indexed limits (resp. colimits) for all \(k \in K\), and that the functor \(C_k \to C_{k'}\) preserves \(I\)-indexed limits (resp. colimits) for every morphism \(k \to k'\) in \(K\). Then the limit \(C:= \lim _{k \in K} C_k\) in \(\Cat _{\infty }\) also admits \(I\)-indexed limits (resp. colimits) and each functor \(C \to C_k\) preserves them.