Subadditivity of the multiplicity in a composition series? Let $\mathsf{C}$ be an abelian category in which every object has finite length. Let $S$ be a simple object in $\mathsf{C}$. For any object $X \in \mathsf{C}$ denote by $[X\colon S]$ the number of times $S$ is a composition factor of $X$. This number is well-defined by the Jordan-Hölder theorem.
Let $\phi,\psi \colon X \rightarrow Y$ be two morphisms in $\mathsf{C}$. Is the following inequality true?
$$[\operatorname{im}(\phi +\psi)\colon S] \leq [\operatorname{im}(\phi)\colon S]+[\operatorname{im}(\psi)\colon S]$$
I know that $\operatorname{im}(\phi), \operatorname{im}(\psi)$ and $\operatorname{im}(\phi+\psi)$ are subobjects of $\operatorname{im}(\phi)+ \operatorname{im}(\psi)$. This might be helpful but I don't know how.
 A: The following certainly works for modules, and should therefore also work in $\mathsf{C}$ by the Freyd–Mitchell embedding theorem:
\begin{align*}
  \newcommand{\im}{\mathrm{im}}
  [\im(ϕ + ψ) : S]
  &≤ [\im(ϕ) + \im(ψ) : S] \\
  &≤ [\im(ϕ) ⊕ \im(ψ) : S] \\
  &= [\im(ϕ) : S] + [\im(ψ) : S] \,.
\end{align*}

Let us check which properties of finite length objects we rely on.
We have the following general result:

*

*If $0 \to X \to Y \to Z \to 0$ is a short exact sequence in $\mathsf{C}$, then $Y$ is of finite length if and only if both $X$ and $Z$ are of finite length, and then
$$
  [Y : S] = [X : S] + [Z : S] \,.
$$
We need the following special cases of this general result:

*

*For any two objects $X$ and $Y$ of finite lengths their direct sum $X ⊕ Y$ is again of finite length, with
$$
  [X ⊕ Y : S] = [X : S] + [Y : S] \,.
$$


*If $X$ is a subobject of an object $Y$ of finite length, then $X$ is again of finite length, with
$$
  [X : S] ≤ [Y : S] \,.
$$


*If $Z$ is a quotient object of an object $Y$ of finite length, then $Z$ is again of finite length, with
$$
  [Z : S] ≤ [Y : S] \,.
$$
Finally, we need the following observations regarding $\im(ϕ) + \im(ψ)$:

*

*We have an epimorphism $\im(ϕ) ⊕ \im(ψ) \to \im(ϕ) + \im(ψ)$, which is induced from the two canonical morphisms $\im(ϕ) \to \im(ϕ) + \im(ψ)$ and $\im(ψ) \to \im(ϕ) + \im(ψ)$.


*We have an inclusion (of subobjects of $Y$) $\im(ϕ + ψ) \to \im(ϕ) + \im(ψ)$.
