Corollary 9.5.3.8 (06Q8)

Corollary 9.5.3.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$: The morphism $U$ is a presentable cocartesian fibration.
$(2)$: The morphism $U$ is a cocartesian fibration and the covariant transport representation of $U$ takes values in the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{LPr}} \subseteq \operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ of Construction 9.5.3.1.
$(3)$: The morphism $U$ is a presentable cartesian fibration.
$(4)$: The morphism $U$ is a cartesian fibration and the contravariant transport representation of $U$ factors through the subcategory $\operatorname{\mathcal{QC}}^{\operatorname{RPr}} \subseteq \operatorname{\mathcal{QC}}^{\operatorname{Acc}}$ of Variant 9.5.3.2.

Proof. The equivalence $(1) \Leftrightarrow (2)$ follows immediately from the definition of presentable fibration and the equivalence $(3) \Leftrightarrow (4)$ follows from the reformulation provided by Proposition 9.5.3.7. Since every presentable fibration is both locally cartesian and locally cocartesian (Proposition 9.5.3.7), the equivalence $(1) \Leftrightarrow (3)$ is a special case of Proposition 6.2.5.7. $\square$