The Jordan Holder theorem for abelian categories states that if you have an object with a "Jordan-Holder Filtration" which is one where the subsequent quotients $X_i/X_{i-1}$ are simple objects, then one can extend any filtration of this object into a Jordan-Holder Filtration. Moreover any two Jordan Holder filtrations have the same set up to multiplicities of simples appearing as quotients as above up to isomorphism.
I can't figure out how to prove this as the proof of the same theorem in the group case doesn't pull through. Does anybody know a reference for this or how to go about proving this?
Note, that if $\mathcal{A}$ is an abelian category and $a\in\text{Obj}(\mathcal{A})$, then $\text{Sub}_{\mathcal{A}}(a)$ is a bounded modular lattice. Now the desired statements are special cases of the Schreier refinement theorem and the Jordan-Hölder theorem for bounded modular lattices (see, for example: Bo Stenström, "Rings of quotients", p.66).