Corollary 9.4.4.8. Let $\kappa $ be a small uncountable regular cardinal and suppose we are given a tower of $\infty $-categories
Assume that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{n}$ is $\kappa $-compactly generated and admits small filtered colimits and that each of the functors $U_{n}: \operatorname{\mathcal{C}}_{n+1} \rightarrow \operatorname{\mathcal{C}}_{n}$ is a finitary isofibration which preserves $\kappa $-compact objects. Then the limit $\varprojlim ( \operatorname{\mathcal{C}}_{n} )$ is $\kappa $-compactly generated. Moreover, an object $X = \{ X_ n \} _{n \geq 0} \in \varprojlim ( \operatorname{\mathcal{C}}_{n} )$ is $\kappa $-compact if and only if each $X_ n$ is a $\kappa $-compact object of $\operatorname{\mathcal{C}}_{n}$.