I'm new to this area, so it may very well be possible that I may be missing something easy here.
Let $E$ be a stable complex vector bundle over $X$ of degree $d$ and rank $n$. Then the moduli space $\mathcal{M}^s(n,d)$ of degree $d$ rank $n$ stable vector bundles can be identified with the space of holomorphic structures on $E$.
Further the tangent space to $\mathcal{M}^s(n,d)$ can be seen to be $H^{0,1}(X, End(E))$, where $End(E)$ denotes the endomorphism bundle. Similarly the tangent space to the space of deformations of a complex structure of $E$ can be seen to be $H^{0,1}(E,T^{1,0}E)$.
Given the identification in the previous paragraph, how does one describe the isomorphism between $H^{0,1}(X, End(E))$ and $H^{0,1}(E,T^{1,0}E)$ ?