Abelian subcategory generated by a full subcategory. If $\mathcal{C}$ is a full subcategory of an abelian category $\mathcal{C}'$ to what extent does the abelian subcategory generated by $\mathcal{C}$ depend on the ambient category $\mathcal{C}'$?
Specifically I have the following question in mind:  Suppose I have a category $\mathcal{C}$ which embeds as a full subcategory in two abelian categories $\mathcal{C}'$, $\mathcal{C}''$. Under what circumstances can I conclude that the abelian categories generated by $\mathcal{C}$ are equivalent?
 A: It can depend a lot. For a very specific example, let $C$ consist of a single object with endomorphism ring $\mathbb{Z}$. $C$ embeds into two familiar abelian categories: the category $\text{Ab}$ of abelian groups, and the category $\text{Ab}^{op}$, which by Pontryagin duality can be thought of as the category of compact Hausdorff abelian groups. In the first case, the abelian subcategory generated by $C$ is the category of finitely generated abelian groups, while in the second case it is the opposite, which by Pontryagin duality can be thought of as the category of compact abelian Lie groups. 
These two categories are not equivalent. For example, the category of finitely generated abelian groups admits a generator (namely $\mathbb{Z}$) but does not admit a cogenerator, and hence its opposite admits a cogenerator but does not admit a generator. 
More generally, if $C$ consists of a single object with endomorphism ring $R$, then $C$ embeds as a full subcategory of any category of modules such that some module has endomorphism ring $R$, and as a full subcategory of the opposite of any category of modules such that some module has endomorphism ring $R^{op}$.  The problem with your hypotheses is that you have no control over the relationship between limits and colimits in $C'$ and limits and colimits in $C''$. If you'd only asked for limits or for colimits then I think there would at least be a universal answer, but here I think there is no "free abelian category" functor. 
