Straightening for lax transformations and adjunctions of (, 2) -categories

We prove an unstraightening result for lax transformations between functors from an arbitrary (, 2) -category to that of (, 2) -categories. We apply this to study partially (op) lax and weighted (co) limits, giving fibrational descriptions of such (co) limits for diagrams valued in (, 2) -categories, to characterize adjoints in (, 2) -categories of functors and (op) lax transformations, and to prove a mate correspondence between lax transformations that are componentwise right adjoints and oplax transformations that are componentwise left adjoints, for such transformations among functors between arbitrary (, 2) -categories.