Corollary 7.1.4.22. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of $\infty $-categories. The following conditions are equivalent:
- $(1)$
For each object $D \in \operatorname{\mathcal{D}}$, the $\infty $-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ has an initial object.
- $(2)$
The functor $U$ is a coreflective localization: that is, it admits a fully faithful left adjoint $F: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$.