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.