In Huybrechts and Lehn's book "The Geometry of Moduli Space of Sheaves", a sheaf $E \in Ob(Coh_{d,d-1}(X))$ is polystable if $E \cong \bigoplus E_{i}$ in $Coh_{d,d-1}(X)$, where the sheaves $E_{i}$ are stable in $Coh_{d,d-1}(X)$ and $\mu(E_{i})=\mu(E)$.
Here is a corollary:
A locally free sheaf $E$ on $X$ is polystable in $Coh_{d,d-1}(X)$, if and only if $E \cong \bigoplus E_{i}$ in $Coh(X)$, where the sheaves $E_{i}$ are $\mu$-stable locally free sheaves with $\mu(E_{i})=\mu(E)$
I can deduce $E \cong \bigoplus E_{i}^{\vee \vee}$ in $Coh(X)$ from $E \cong \bigoplus E_{i}$ in $Coh_{d,d-1}(X)$, and since a summand of a locally free sheaf is locally free, $E_{i}^{\vee\vee}$ are locally free sheaves,but they may not be $\mu$-stable. What am I missing?
