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

Abstract: 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.