Let $A$ and $B$ be abelian categories, then $A \times B$ is an abelian category. Also denote $Ab$ to be the category of abelian groups or any abelian category.

If want to study a bi-additive (also called bilinear) functor $$F(-,-): A\times B \rightarrow Ab$$ , which is right exact in each variable, then Deligne's tensor product $A\boxtimes B$ which is an abelian category allows us to study bi-additive functor F on $A\times B$ by studying an additive right exact functor $$\overline{F}:A\boxtimes B \rightarrow Ab$$.

More precisely, we have a functor $\boxtimes: A \times B \rightarrow A\boxtimes B$ such that for any $F$ described above, then there exists a unique additive right exact functor $\overline{F}: A\boxtimes B \rightarrow Ab$ such that $F= \overline{F} \circ \boxtimes$.

I have a few questions regarding this construction.

1) Is $\boxtimes$ essentially surjective? (put any conditions if you want)

2) For $A$= the oppositive category of finite typed commutative group schemes over a field $k$ and $B$= the category of finite typed commutative group schemes over $k$. Does $A \boxtimes B$ exist? If yes, can we describe its objects if $\boxtimes$ is not essentially surjective?

3) Deligne's tensor product allows us to deal with bi-additive functor that is right exact in each variable as mentioned. Does there exist an alternative constuction allowing us to deal with a bi-additive functor that is left exact in each variable?