I'm looking for a notion of an Abelian category $\mathcal{A}$ "generated" by a given category $\mathcal{C}$


More precisely I need something along the following lines. Denote $\mathcal{Ab}\_2$ the 2-category of Abelian categories and $\mathcal{Cat}$ the 2-category of categories. We have the forgetful 2-functor $\mathcal{F}: \mathcal{Ab}\_2 \rightarrow \mathcal{Cat}$. Is there an adjoint 2-functor $\mathcal{G}: \mathcal{Cat} \rightarrow \mathcal{Ab}\_2$ ?


I suspect the answer is "yes" because it can be constructed along the following lines. Denote $\mathcal{Ab}$ the category of Abelian groups. For any category $\mathcal{C}$, the category $\mathcal{Hom(C, Ab)}$ is Abelian. Moreover, we have the natural functor $\mathcal{i:C \rightarrow Hom(Hom(C,Ab),Ab)}$. Thus $\mathcal{C}$ is embedded in the Abelian category $\mathcal{D:=Hom(Hom(C,Ab),Ab)}$ and we can take the Abelian category generated by $\mathcal{C}$ within $\mathcal{D}$. The result is supposed to be $\mathcal{G(C)}$


However, the only construction I managed to search up is the Karoubi envelope which generates a pseudo-Abelian category. So either my purported construction is wrong or simply not popular. Which is it?


EDIT: I realized my construction amounts to $\mathcal{G(C):=Hom(C,Ab^{op})}$. At least for small $\mathcal{C}$ this is indeed adjoint to $\mathcal{F}$, provided we interpret $\mathcal{Ab\_2}$ as having *right* exact functors for 1-morphisms. Here $\mathcal{C}$ embeds by applying opposite Yoneda and taking the freely generated Abelian group.



Is there an "Abelian envelope" 2-functor?
I'd be inclined to go in a slightly different direction in constructing the free abelian category generated by a category. I would do it in stages: 


* First construct the free $Ab$-enriched category $F\_1(C)$ generated by a category $C$. This would have the same objects as $C$, but the hom $F\_1(C)(a, b)$ between two objects would be the free abelian group generated by the hom $C(a, b)$ in $C$.
* Next, to any $Ab$-enriched category $C$, we may freely adjoin finite limits. We can do this by embedding $C$ into $(Ab^C)^{op}$ by the dual of the Yoneda embedding, and then cutting down to the full subcategory $F\_2(C)$ of $Ab$-enriched functors which arise as limits of finite diagrams of opposites of representables $C(c, -)^{op}$ in $(Ab^C)^{op}$.
* Next, to any finitely complete $Ab$-enriched category $C$, pass to the so-called [ex/lex completion](http://ncatlab.org/nlab/show/regular+and+exact+completions#the_exlex_completion_3) $F\_3(C)$. The "lex" here refers to left exactness (i.e., closed under finite limits, and functors preserving such), whereas the "ex" refers to [Barr exactness](http://ncatlab.org/nlab/show/exact+category) and functors preserving such.


The point is that an abelian category is precisely a Barr-exact (finitely complete) $Ab$-enriched category (see Freyd-Scedrov's Categories, Allegories, page 90, where a Barr-exact category is what they call an effective regular category). So the free abelian category generated by a category $C$ is $(F\_3 \circ F\_2 \circ F\_1)(C)$. I did not check carefully that the ex/lex completion, as described in the nLab article, lifts from the $Set$-enriched world to the $Ab$-enriched world, but I think it's okay. 


Not yet sure how this relates to your proposed construction (there are some size considerations to, er, consider), but I think the answer to your first question is 'yes'. 


**Addendum:** In response to Martin's comment, and as confirmation that the description above is correct, here is a [paper](http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=4940460) by Carboni and Magno which in the course of things constructs the free abelian category generated by an additive category: see page 300. The free abelian category on the terminal category $\mathbf{1}$ is $C\_{ex}$, where $C$ is the category opposite to that of finitely generated abelian groups. There may be simpler descriptions, but I'm not prepared to say more on that now. 


