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

Lax monoidal adjunctions, two-variable fibrations and the calculus of mates