نبذة مختصرة : 51 pages, v2. ; We provide a calculus of mates for functors to the $\infty$-category of $\infty$-categories. As the most important application we show that given an adjunction between symmetric monoidal $\infty$-categories, there is an equivalence between lax symmetric monoidal structures on the right adjoint and oplax symmetric monoidal structures on the left adjoint functor. 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 relate to functors out of the Gray tensor product of $(\infty, 2)$-categories.
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