Kerodon

Proposition 9.2.8.1. Let $\kappa \leq \lambda $ be regular cardinals, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a right fibration of $\infty $-categories, and let $X$ be an object of $\operatorname{\mathcal{E}}$. Assume that $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{E}}$ are $(\kappa ,\lambda )$-cocomplete and that the functor $U$ is $(\kappa ,\lambda )$-finitary. If $U(X)$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{C}}$, then $X$ is a $(\kappa ,\lambda )$-compact object of $\operatorname{\mathcal{E}}$.