Cannonical evaluation map Let $C$ be a curve over $\mathbb{C}$, and $E$ be a vector bundle on $C$ such
that $H^0 (C, E) \neq 0$. Everyone talks of the evaluation map 
$H^0 (C, E)\otimes O_C\longrightarrow E$. What is this map exactly? It is generally described as $s\longrightarrow s_x$ that is restricting a global section to it's stalk. 
But what is the sheaf level map? Is it given by $H^0 (C, E)\otimes O_C(U)\longrightarrow E(U)$, sending $s\otimes t\mapsto t.s|_U$? 
Or does it mean taking as many copies of $O_C$ as there are global sections? I saw somewhere that if this map is surjective then $E$ is generated by global sections. How is that?
Any help will be appreciated! Thanks!
 A: 1) For  the locally free sheaf $\mathcal E$ the definition of the evaluation map $ H^0 (C, \mathcal E)\otimes_\mathbb C O_C\longrightarrow \mathcal E$ is indeed given on the presheaf level by 
$$H^0 (C, \mathcal E)\otimes _\mathbb C O_C(U)\longrightarrow  H^0 (U, \mathcal E):s\otimes f \mapsto f.s|_U$$
2) We say that the vector bundle $E$ associated to the locally free sheaf $\mathcal E$ is generated by its global sections if one of the following equivalent conditions is satisfied:     
a) For every $x\in C$ the stalk  $\mathcal E_x$ seen as an $\mathcal O_x$-module  is generated by the global sections $ H^0(C, \mathcal E)$.
b) For every $x\in C$ and every element $e\in E(x)=\mathcal E_x/\mathfrak m_x\mathcal E_x$ in the fiber   of  $E$ over $x$ there exists a global section $s\in H^0(C, E)$ with $s(x)=e$.  
The equivalence of these conditions is proved by applying the NAKAYAMA lemma, and in my opinion displays the true significance of that lemma.
Notice that what I say applies to a vastly more general situation: coherent sheaves over a locally ringed space.
