As far as I know there are different ways to categorify the notion of vector space/module. These appear (for example) when trying to find extended TQFTs. There are at least two ways (presented at $2$-vector-space at nLab and more generally in the article on $(\infty,n)$-modules):

- The 2-category $\mathsf{Vect}_2$ whose objects are $k$-linear categories, $1$-morphisms are functors and $2$-morphims are natural transformations;
- The 2-category $1\text{-}\mathsf{Alg}$ whose objects are associative $k$-algebras, morphisms are bimodules and $2$-morphisms are bimodules morphisms.

The two are linked by some kind of Tannaka duality: there's a functor $1\text{-}\mathsf{Alg} \to \mathsf{Vect}_2$ (described here) that maps an algebra $A$ to its category of modules $\mathsf{RMod}_A$, an $(A,B)$ bimodule $N$ to the functor $- \otimes_A N$, and a bimodule morphism to the associated natural transformation.

I think this is fully faithful but not essentially surjective. Am I right? How to describe the essential image of this functor? A category in the image has to at least be monoidal and rigid, for example. Is the image possibly the fully dualizable objects?