On local fibrations of (∞,2)-categories

In this work we provide a model-independent notion of local fibrations of (∞,2)-categories which generalises the well-known theory of locally coCartesian fibrations of (∞,1)-categories. Based on previous work, we construct a model category which serves as a specific combinatorial model for this type of fibrations. 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 (∞,2)-categories between the (∞,2)-category of $(0,1)$-fibrations over S (also known as inner coCartesian fibrations) and the (∞,2)-category of functors S → C\!at(∞,2) with values in (∞,2)-categories. Given an (∞,2)-category B, our Grothendieck construction can be specialised to produce an equivalence between the (∞,2)-category of local fibrations over B and the (∞,2)-category of oplax unital functors with values in C\!at(∞,2). Finally, as an application of our results we provide a version of the Yoneda lemma for (∞,2)-categories.