kappa ind object and straightening

Let \(\mathcal{C}\) be a small infinity category, \(\mathcal{P}(\mathcal{C})\) the infinity presheaf category, \(\kappa\) a regular cardinal and \(F \in \mathcal{P}(\mathcal{C})\) an infinity presheaf.

TFAE:

1. \(\mathcal{F} \in \mathrm{Ind}_{\kappa}\)
   
2. the unstraightening yields a right fibration \(\mathrm{Un}(\mathcal{F}): \mathcal{E} \rightarrow \mathcal{C}\) where \(\mathcal{E}\) is \(\kappa\)-kappa filtered infinity category
   

FW: 6.59 + 6.61