  We provide a calculus of mates for functors to the $\infty$-category of
$\infty$-categories and extend Lurie's unstraightening equivalences to show
that (op)lax natural transformations correspond to maps of (co)cartesian
fibrations that do not necessarily preserve (co)cartesian edges. As a sample
application we obtain an equivalence between lax symmetric monoidal structures
on right adjoint functors and oplax symmetric monoidal structures on the left
adjoint functors between symmetric monoidal $\infty$-categories that is
compatible with both horizontal and vertical composition of such structures.
  As the technical heart of the paper we study various new types of fibrations
over a product of two $\infty$-categories. In particular, we show how they can
be dualised over one of the two factors and how they encode functors out of the
Gray tensor product of $(\infty, 2)$-categories.
