This can be seen as a follow up my question here:


[Is there a notion of "fibered category with boxproducts"?](https://mathoverflow.net/questions/28152/is-there-a-notion-of-fibered-category-with-boxproducts)


Given a monoidal fibration $f:E\rightarrow B$ 
(i.e. a strict monoidal functor between monoidal categories which is a fibration of ordianary categories)
where the base is a cartesian monoidal category endowed with a grothendieck topology. 
What are the right conditions for such a fibration be called a stack?


I guess it is not enough to ask that $E(X)\rightarrow Desc(X,U)$ is an equivalence of ordinary categories. Insted one should need some further condition that ensures the following:


"if $(\phi\_i)$ can be glued to $\phi$ and $(\psi\_j)$ can be glued to $\psi$ than $(\phi\_i\boxtimes \psi\_j)$ can be glued to $\phi\boxtimes \psi$"


Does this notion exist yet? What would be the right condition? 


Examples I have in mind are 


$B$=geometric objects for example smooth varieties and


$E$=sheaves for example $\mathcal{D}\_X$-modules



Notion of stack fibered in monoidal categories?
I would take the view point that a monoidal category is a bicategory with one object. (Then a category fibered in monoidal categories should be the same thing as a weak functor into bicategories that "only hits monoidal categories".) In other words, what you should have is that this fibration is a 2-stack when viewed as a fibration in bicategories. The descent condition should then be that the canonical map $E|\_X \to Desc(X,U)$ be an equivalence of bicategories, where each of these monoidal categories is viewed as a bicategory, which is equivalent to Jeff's comment; this is just saying that $E|\_X \to Desc(X,U)$ is a monoidal equivalence.


