Proposition 9.4.8.1. Let $\kappa $ be an uncountable regular cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a flat inner fibration of simplicial sets which is essentially $\kappa $-small. Then:
- $(1)$
The projection map $V: \operatorname{Fun}(\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } ) \rightarrow \operatorname{\mathcal{C}}$ is a cocartesian fibration of simplicial sets (see Construction 4.5.9.1).
- $(2)$
Let $V^{\dagger }: \operatorname{Fun}( \operatorname{\mathcal{E}}/ \operatorname{\mathcal{C}}, \operatorname{\mathcal{S}}^{< \kappa } )^{\dagger } \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ be a cartesian conjugate of $V$ (Definition 8.6.1.1). Then $V^{\dagger }$ is a fiberwise $\kappa $-cocompletion of $U^{\operatorname{op}}: \operatorname{\mathcal{E}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$.