Let $R$ be a commutative ring. A categorification of the concept of an $R$-module might be a $R$-linear cocomplete category (there are some reasons for this). For $R$-modules, we can define their tensor product as a classifying object of the $R$-bilinear maps on the product and it exists. My question is now if we can do the same thing for $R$-linear cocomplete categories.

**Question.** Let $C,D,E$ be $R$-linear cocomplete categories. A functor $C \times D \to E$ is called $R$-bilinear if it is cocontinuous and $R$-linear in each variable. We get a $2$-functor $\text{Bilin}(C \times D,-)$ from $R$-linear cocomplete categories to categories. Is it $2$-representable?

I've already tried to imitate the usual construction of the tensor product of $R$-modules, but the details do not work out. I'm pretty sure that it does work somehow and I would be glad if someone could explain this in detail.