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 duduce $E \cong \bigoplus E_{i}^{\vee \vee}$ 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?

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?