(Def 1) A surjective morphism of sheaves (on a topological space $X$) is usually (and most naturally) defined in the algebraic geometry literature as a morphism of sheaves such that $\mathcal{F} \to \mathcal{G} \to 0$ is exact (where a morphism of sheaves is just a morphism of presheaves).
(Def 2) The definition of exactness in an abelian category is that $A\xrightarrow{\alpha}B\xrightarrow{\beta}C$ is exact iff $\operatorname{ker}\beta = \operatorname{Im}\alpha.$
(Def 3) The image in the category of sheaves is the sheafification of the image presheaf.
(Fact i) In general, one may prove that a sequence of sheaves $0 \to \mathcal{J} \to \mathcal{F} \to \mathcal{G} \to 0$ is exact if and only if for every $x \in X$ the sequence on stalks $0 \to \mathcal{J}_x \to \mathcal{F}_x \to \mathcal{G}_x \to 0$ is exact.
(Fact ii) On the other hand, a common warning is that there are maps of sheaves such that $\mathcal{F}_x \to \mathcal{G}_x$ is surjective for every $x \in X$ but there exists an $U \subset X$ such that $\mathcal{F}_U \to \mathcal{G}_U$ is not surjective.
Putting (i)+(ii) together one gets that:
There exists morphism of sheaves such that $\mathcal{F} \to \mathcal{G} \to 0$ is exact but $\mathcal{F}_U \to \mathcal{G}_U \to 0$ is not exact for some $U.$
Since limits and colimits of presheaves are defined pointwise, this means the following:
Calling $\operatorname{Im}\alpha$ the image presheaf and $\operatorname{Im}^+$ the sheafification, there exists morphisms of sheaves such that $\mathcal{F} \to \mathcal{G} \to 0$ is exact in the category of sheaves but not exact in the category of presheaves.
This makes sense, since the inclusion $\operatorname{Sh}(X) \hookrightarrow \operatorname{Psh}(X)$, is a right adjoint to sheafification, so it preserves limits but not necessarily colimits, hence left exact sequences of sheaves give left exact sequences of presheaves but this does not happen for right exactness. I think the failure to preserve right exactness also boils down to the difference between $\operatorname{Im}$ and $\operatorname{Im}^+.$
Question
Is what I said above correct?
If so, can someone give me an example of a right exact sequence $\mathcal{F} \to \mathcal{G} \to 0$ of sheaves which is not right exact as a sequence of presheaves?