Proposition 8.6.8.13. Let $\kappa $ be an uncountable cardinal, let $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ be a cartesian fibration of simplicial sets which is essentially $\kappa $-small. Set $\operatorname{\mathcal{E}}= \operatorname{\mathcal{C}}\times _{ \operatorname{\mathcal{C}}^{\triangleright } } \overline{\operatorname{\mathcal{E}}}$, so that $\overline{U}$ restricts to a cartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. The following conditions are equivalent:
- $(1)$
The contravariant transport representation $\operatorname{Tr}_{ \overline{\operatorname{\mathcal{E}}} / \operatorname{\mathcal{C}}^{\triangleright } }: ( \operatorname{\mathcal{C}}^{\triangleright } )^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}^{< \kappa }$ is a limit diagram.
- $(2)$
The restriction functor
\[ \operatorname{Fun}^{\operatorname{Cart}}_{ / \operatorname{\mathcal{C}}^{\triangleright } }( \operatorname{\mathcal{C}}^{\triangleright }, \overline{\operatorname{\mathcal{E}}} ) \rightarrow \operatorname{Fun}^{\operatorname{Cart}}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}) \]is an equivalence of $\infty $-categories.