I am currently reading chapter 6.5 (Morphisms of Sheaves and Sheafification) in Bosch: Algebraic Geometry and Commutative Algebra and there is something in the context of epimorphisms of sheaves that confuses me.
Definition. (Bosch) Let $\varphi \colon \mathcal{F} \rightarrow \mathcal{G}$ be a morphism of sheaves. Then $\varphi$ is called an epimorphism, if $\operatorname{im} \varphi = \mathcal{G}$.
In the chapter, $\operatorname{im} \varphi$ is defined as the sheafification of the presheaf $$ \begin{align}(\operatorname{im} \varphi)_{\text{pre}} \colon \mathbf{Ouv(X)} &\longrightarrow \mathbf{Ring} \\ U &\longmapsto \operatorname{im} \varphi(U) \end{align} $$
The sheafification of a presheaf $\mathcal{H}$ is defined by a universal property and constructed via the associated presheaf $\mathcal{H}^+$, where $$ \mathcal{H}^+(U) := \varinjlim_{\mathfrak{U} \in I_U} H^0(\mathfrak{U}, \mathcal{H}).$$
Then we get a sheafification $\mathcal{H}' = \mathcal{H}^{++}$. But in our case one can even show $$(\operatorname{im} \varphi)_{\text{pre}}^{'} = (\operatorname{im} \varphi)_{\text{pre}}^{+}.$$
Here is where I am struggling:
I claimed and tried to proof the following
Lemma. Let $\varphi \colon \mathcal{F} \rightarrow \mathcal{G}$ be a morphism of sheaves. Then $\varphi$ is an epimorphism if and only if $\varphi$ is locally surjective (i.e. for any section $g \in \mathcal{G}(U)$ and any $x \in U$ there is an open neighbourhood $U_x \subseteq U $ of $x$ and a section $f \in \mathcal{F}(U_x)$ with $\varphi(U_x)(f) = g\vert_{U_x}$).
I did several attempts involving the mentioned direct limit and it always nearly worked out, but so far no success. Is my claimed Lemma even true? Is the definition of epimorphism a correct one? Or should it demand an isomorphism $\operatorname{im} \varphi \cong \mathcal{G}$ instead of $\operatorname{im}\varphi = \mathcal{G}$ since the sheafification of $ (\operatorname{im}\varphi)_{\text{pre}} $is only unique up to isomorphisms anyway? Should I (therefore also) alter the definition of locally surjective? But how? I am confused. Any help to clear things up is appreciated. :-)
N.B.: I am aware of the equivalence
Let $\varphi \colon \mathcal{F} \rightarrow \mathcal{G}$ be a morphism of sheaves. Then $$ \varphi \text{ is locally surjective} \quad \iff \quad \varphi_x \colon \mathcal{F}_x \rightarrow \mathcal{G}_x \text{ is surjective for all } x. $$
But I would prefer not to use it.