TITLE: Higher vector spaces
QUESTION [9 upvotes]: 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?

REPLY [7 votes]: As I already noted in the comments: A $k$-linear category is equivalent to a module category iff it is abelian, cocomplete, and there exists a compact projective generator. This is a version of Morita's theorem, which can be found e.g. in this article by Bernhard Keller. 
For the question about which functors are equivalent to tensor functors the answer is given by Eilenberg-Watts theorem stating that these are exactly the functors which are right exact and preserve small coproducts (see e.g. Athens following article in nlab: http://ncatlab.org/nlab/show/Eilenberg-Watts+theorem )