Straightening for lax transformations and adjunctions of $(\mathrm{\infty },2)$-categories

We prove an unstraightening result for lax transformations between functors from an arbitrary $(\mathrm{\infty },2)$-category to that of $(\mathrm{\infty },2)$-categories. We apply this to study partially (op)lax and weighted (co)limits, giving fibrational descriptions of such (co)limits for diagrams valued in $(\mathrm{\infty },2)$-categories, to characterize adjoints in $(\mathrm{\infty },2)$-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 $(\mathrm{\infty },2)$-categories.