I have some difficulty understanding the above proof.
Here $C(\mu)$ denotes the category of semi-stable vector bundles (locally free sheaf) of slope $\mu$. Here the sheaves are considered over a smooth connected algebraic curve $X$.
A locally free sheaf $F$ is called semi-stable if for every subsheaf $G$, the slope of $G$, $\mu(G)$, is less than equal to slope of $F$, $\mu(F)$. The slope of a locally free sheaf is defined as $deg(F)/rank(F)$.
I've two questions about the above proof
It says that the $ker(f)$ and $coker(f)$ are locally-free sheaves. Well, the $ker(f)$ is a subsheaf of $E$, which is a locally-free sheaf, so $ker(f)$ is also a locally free sheaf ( because these are sheaves on a smooth curve and therefore stalks are pid, and a submodule of a free module over a pid is again a free module). But I've absolutely no idea how it says $coker(f)$ is also a locally-free sheaf. Why is this true?
In the proof of $ker(f)$ is semistable, it says if it's not semistable it would have a sub-bundle of slope of slope greater than $\mu$. So, somehow there are assuming slope of $ker(f)$ is also $\mu$. So, I guess since slope of $E$ and $F$ is $\mu$ and they are semistable then it is forcing the slope of both $ker(f)$ and $coker(f)$ to be $\mu$. Is this correct?