Let $E$ be a vector bundle over a compact Riemann surface $X$, and let $$0=E_0\subsetneq E_1\subsetneq \ldots \subsetneq E_n=E$$ be its Harder-Narasimhan filtration: we have $V_i:=E_i/E_{i-1}$ semistable and $\mu_i:=\mu(V_i) > \mu_{i+1}$ for all $i>0$. My question is how to construct the Harder-Narasimhan filtration for $End(E)$ from there. Here is what I have from Atiyah-Bott "Yang-Mills equations over Riemann surfaces" (p.590). Setting $W_i:=\{\phi \in End(E)~|~\phi(E_k)\subset E_{k+i}~ \forall k\}$, one obtains a filtration \begin{equation}(*)\quad \quad 0=W_{-n}\subsetneq \ldots \subsetneq W_0\subsetneq \ldots \subsetneq W_{n-1}=End(E).\end{equation} We have that $W_{i}/W_{i-1}=\bigoplus_k Hom(V_k,V_{k+i})$ is a direct sum of semistable vector bundles of respective weights $\mu_{k+i}-\mu_k$.

The paper concludes that

- Any subbundle $0\neq F\subset End(E)/W_0$ satisfies $\mu(F)<0$.
- The Harder-Narasimhan filtration for $End(E)$ is a refinement of $(*)$.

Could anybody help me understand this conclusion ? I see that $W_0/W_{-1}$ is semistable of slope $0$ and that we may refine the filtration $(*)$ using Harder-Narasimhan for $W_i/W_{i-1}$ to obtain semistable successive quotients, but then the weights will not necessarily decrease.