Warning 9.2.9.17. In the formulation of Proposition 9.2.9.16, we generally cannot replace $(\ast )$ by the following weaker condition:
- $(\ast _0)$
For each vertex $C \in \operatorname{\mathcal{C}}$, the image $F(C)$ is $(\kappa ,\lambda )$-compact when viewed as an object of the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
However, condition $(\ast _0)$ does imply condition $(\ast )$ under either of the following additional assumptions:
The inner fibration $U$ is locally cartesian. Moreover, for every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the contravariant transport functor $e^{\ast }: \operatorname{\mathcal{E}}_{C'} \rightarrow \operatorname{\mathcal{E}}_{C}$ is $(\kappa ,\lambda )$-finitary.
For every edge $e: C \rightarrow C'$ of $\operatorname{\mathcal{C}}$, the image $F(C) \rightarrow F(C')$ is a locally $U$-cocartesian edge of $\operatorname{\mathcal{E}}$.
See Warning 9.2.9.15.