Relative Monoidal Bondal-Orlov

In this article we study a relative monoidal version of the Bondal-Orlov reconstruction theorem. We establish an uniqueness result for tensor triangulated category structures $(\boxtimes,\mathbb{1})$ on the derived category $D^{b}(X)$ of a variety $X$ which is smooth projective and faithfully flat over a quasi-compact quasi-separated base scheme $S$ in the case where the fibers $X_{s}$ over any point $s\in S$ all have ample (anti-)canonical bundles. To do so we construct a stack $\Gamma$ of dg-bifunctors which parametrize the local homotopical behaviour of $\boxtimes$, and we study some of its properties around the derived categories of the fibers $X_{s}$.