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](http://ncatlab.org/nlab/show/2-vector+space) 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](http://ncatlab.org/nlab/show/%28infinity%2Cn%29-module)) 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?



Higher vector spaces
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](http://webusers.imj-prg.fr/~bernhard.keller/publ/cam.pdf). 
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> ) 


