On local fibrations of (∞,2)-categories

We provide a model independent notion of local fibrations of .1;2/-categories which generalises the wellknown theory of locally cocartesian fibrations of .1;1/-categories.Based on previous work, we construct a model structure which serves as a specific combinatorial model for this type of fibration.Our main result is a generalisation of the locally cocartesian straightening and unstraightening construction of Lurie which yields for any scaled simplicial set S an equivalence of .1;2/-categories between the .1;2/-category ¿ib 0;1 .S / of .0;1/-fibrations over S and the .1;2/-category of functors Fun.S; »icat 1 /.Given an .1;2/-category » our Grothendieck construction can be specialised to produce an equivalence between the .1;2/-category of local fibrations over » and the .1;2/-category of oplax normalised functors with values in »icat 1 .Finally, as an application of our results we provide a version of the Yoneda lemma for .1;2/-categories.