Question

Is there a notion of fibered category with box products? By this I roughly mean a fibration $C\rightarrow B$ where $B$ has finite products, along with functors $$\boxtimes: C(X)\times C(Y)\rightarrow C(X\times Y)$$ and some coherent isomorphisms, for example:

$$(f\times g)^* (M\boxtimes N) \leftrightarrow (f^* M) \boxtimes (g^* N)$$ and $$(M\boxtimes N)\boxtimes L \leftrightarrow M\boxtimes (N \boxtimes L) $$

This situation occurs often in geometry, for example:

B=Varieties, C=quasi coherent sheaves, D-modules

Answer 1

Yes, there is. In this paper I called it a monoidal fibration (with cartesian monoidal base), but I'm sure that other people had thought about it before. There are some nice things you can say especially in the case when the base is cartesian; for instance you automatically get a monoidal structure on each fiber defined by $M\otimes N = \Delta^*(M\boxtimes N)$, and you can recover the box-product from the fiberwise monoidal structure via $M\boxtimes N = \mathrm{pr}_1^* M \otimes \mathrm{pr}_2^* N$.