Why is every sheaf of $\mathcal{O}_{X}$-modules not generated by global sections?

Let $(X, \mathcal{O}_{X})$ be a ringed-space. Let $\mathcal{F}$ be a sheaf of $\mathcal{O}_{X}$-modules. For any section $s \in \Gamma(X, \mathcal{F})$, we define a morphism, $$\mathcal{O}_{X} \longrightarrow \mathcal{F}$$ by $f \mapsto f \cdot s|_{U}$ for every open set $U \subseteq X$ with $f \in \mathcal{O}_{X}(U)$. Conversely, any morphism of sheaves $\varphi: \mathcal{O}_{X} \rightarrow X$ as above can be realized in this way by choosing the section $\varphi(1)$. Now if $I$ is any indexing set, any morphism $$\phi: \mathcal{O}_{X}^{(I)} \longrightarrow \mathcal{F}$$ is determined by a component family, $$\{ \varphi_{i}: \mathcal{O}_{X} \longrightarrow \mathcal{F} \}_{i \in I}$$ In turn, as described above, we then get sections $s_{i} = \varphi_{i}(1)$ which determine each of these morphisms. But since $\mathcal{O}_{X}$ is a generator in the category of sheaves of $\mathcal{O}_{X}$-modules, we can always find an index set $I$ so that there is a surjection, $$\phi: \mathcal{O}_{X}^{(I)} \longrightarrow \mathcal{F}$$ as above. But then we could choose out sections like we just did and suddenly we have that $\mathcal{F}$ is generated by those global sections.

Where is the flaw in this reasoning? I know it is obviously wrong because we could find $\mathcal{F}$ having no non-trivial global sections at all, despite being a non-zero sheaf.

• "But since $\mathcal{O}_X$ ia a generator in the category of sheaves of $\mathcal{O}_X$-modules"? – Lord Shark the Unknown Feb 10 '18 at 17:04
• @LordSharktheUnknown are you saying this is untrue, or just not sure of what I mean by this statement? – Luke Feb 10 '18 at 17:08

As suggested in the comment by Lord Shark the Unknown, $\mathcal{O}_X$ is not a generator of the category of sheaves of $\mathcal{O}_X$-modules though it has a generator, namely $\bigoplus_{U: \text{ open}} \mathcal{O}_U$.
• So I've had another look at how generators are defined in Tohoku and you are right, that was a stupid mistake on my part. But even putting that aside, given a sheaf of $\mathcal{O}_{X}$-modules, can you not always find a surjective morphism from some direct sum of copies $\mathcal{O}_{X}$ onto $\mathcal{F}$? – Luke Feb 10 '18 at 17:28