The straightening-unstraightening equivalence between presheaves and fibrations lifts to a biequivalence of infinity-cosmoi for all models of (infinity,1)-categories.
Quasi-categories vs S egal spaces
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Cosmological Unstraightening
The straightening-unstraightening equivalence between presheaves and fibrations lifts to a biequivalence of infinity-cosmoi for all models of (infinity,1)-categories.