In addition to the Jordan-Holder theorem for groups, there are various Jordan-Holder Theorems for other categories:

1. Finite dimensional representations have filtrations whose associated graded consists of irreducible representations. Any other such associated graded is the same up to permutation of its elements.

2. Artinian modules have filtrations whose associated graded consists of simple modules. Any other such associated graded is the same up to permutation of its elements.

3. Finitely generated $A$-modules have filtrations whose associated graded consists of modules of the form $A / p$ for a prime $p$. Any other such associated graded is the same up to permutation of its elements.

There is a commonality to the proofs of these as well. I am wondering if someone has come up with a categorical version of the Jordan-Holder theorem, which in a sense encompasses these ones.

For instance, one might try to look at the category of subquotients $\text{SubQuot}(X)$ of an object $X$ in a category $C$. (I haven't checked this very well, so be cautious). Objects in this category are equivalence classes of pairs $(Z, Y)$ with $Y \in \text{Quot}(X)$ and $Z \in \text{Sub}(X)$, where two such pairs $(Y, Z)$ and $(Y', Z')$ are considered equivalent when $Y' \cong Y$ in $\text{Quot}(X)$ and $Z \rightarrow X \rightarrow Y' \cong Z' \rightarrow X \rightarrow Y'$. Morphisms $f : (Y, Z) \rightarrow (Y', Z')$ are pairs of maps $Y \twoheadrightarrow Y'$ in $\text{Quot}(X)$ and $(\text{im}(Z' \rightarrow X \rightarrow Y') \rightarrow Y') \rightarrow (\text{im}(Z \rightarrow X \rightarrow Y') \rightarrow Y')$ in $\text{Sub}(Y')$.

To construct the coproduct of $(Y, Z)$ and $(Y', Z')$, we simply take the pushout $Y''$ of $Y$ and $Y'$ in $C$ (coproduct in $\text{Quot}(X)$), and the pullback $Z''$ of $Z$ and $Z'$ in $C$ (product in $\text{Sub}(X)$). The coproduct is the pair $(Y'', Z'')$.

A filtration is then a sequences of subquotients $(Y_i, Z_i)$ where $0 \rightarrow Z_i \rightarrow Y_i \rightarrow Y_{i+1} \rightarrow 0$ is exact.

Coproducts can be used to refine filtrations. The Jordan-Holder Theorem for modules then asserts that, for two filtrations with simple subquotients $F$ and $G$ (the subquotients have no nontrivial subquotients), we can take a mutual refinement $F \vee G$. The mutual refinement, after discarding its redundant elements, has the same subquotients as both $F$ and $G$.

I think this can be applied to groups and modules. However, I am particularly interested to see if anyone can make something like this work for the third example above, because I don't quite see how it fits in with the rest.