This is a subentry of (infinity,1)-Grothendieck construction.
which (under an assumption on the parameter ) can be shown to be a Quillen equivalence between the overcategory of simplicial sets equipped with the model structure for right fibrations (also called contravariant model structure in HTT) and the category of simplicial presheaves equipped with global projective model structure.
There is also a Quillen equivalence
Here (in both cases) is called straightening functor and is called unstraightening functor. These names have been chosen due to the fact that objects in the left hand category are defined be existential assertions and choices where on the right side these properties become coherence laws being part of the structure.