Let $X$ be a quasi-projective irreducible scheme, $\mathcal{F}\_1$ a globally generated $\mathcal{O}\_X$-module and $\mathcal{F}\_2$ a coherent sheaf over $X$. Suppose that $\mathcal{F}\_1$ is globally generated by the sections $s\_1,...,s\_n \in \Gamma(X,\mathcal{F}\_1)$. I would imagine that it follows from the definition of morphism of sheaves that any morphism from $\mathcal{F}\_1$ to $\mathcal{F}\_2$ is determined by the image of $s\_1,...,s\_n$ and these sections are mapped to elements in $\Gamma(X,\mathcal{F}\_2)$. Is this true? More precisely, is it true that there exists an **injective morphism** from $\mbox{Hom}\_X(\mathcal{F}\_1,\mathcal{F}\_2)$ to $\mbox{Hom}\_{\mathcal{O}\_X(X)}(\mathcal{F}\_1(X),\mathcal{F}\_2(X))$?



Morphisms between a globally generated sheaf and a coherent sheaf(Edited)
It is not true. Take for example $F\_1$ and $F\_2$ to be the structure sheaves of two distinct points on $X$. Then both are globally generated, the spaces of global sections are 1-dimensional, and there is a nontrivial morphism between these vector spaces. But there is no morphism between the sheaves.


On the other hand, the morphism $Hom(F\_1,F\_2) \to Hom(\Gamma(F\_1),\Gamma(F\_2)$
(the action of the functor of global sections $\Gamma$) is injective if $F\_1$ is globally generated. Indeed, the condition ensures that the map $\Gamma(F\_1)\otimes O\_X \to F\_1$ is surjective, hence applying $Hom(-,F\_2)$ gives an injective morphism 
$$
Hom(F\_1,F\_2) \to Hom(\Gamma(F\_1)\otimes O\_X,F\_2) = Hom(\Gamma(F\_1),\Gamma(F\_2)).
$$


