I am trying to understand the structure of the smallest abelian subcategory of an abelian category that contains one object $X$ and all endomorphisms of that object (or rather containing a finite number of object with all morphisms between, but this seems the same to me by just taking the sum). Just taking the intersection over all subcategories containing my object and morphisms seems a bad idea, since sums, kernels etc are just determined up to isomorphism. I am not bothered about having "too many" isomorphic objects in my subcategory, though. I stumbled over the notion of subquotients. Since kernels, cokernels are subquotients and subquotients of subquotients are subquotients it seems to me as if I could describe the objects of my subcategory the following way: Just take all objects isomorphic to subquotients of finite direct sums of $X$. Does this make sense? Thank you!