Global sections need not preserve surjections

Let $\mathcal{C}$ be the sheaf of continuous real-valued functions on $S^1$, and let $\underset{‾}{\mathbb{Z}}$ be the subsheaf of locally constant integer-valued functions. Let $\mathcal{Q}$ be the sheafification of the presheaf quotient $V\mapsto \mathcal{C}(V)\mathrm{/}\underset{‾}{\mathbb{Z}}(V)$. Every germ of $\mathcal{Q}$ is represented locally by a continuous function, so the canonical map $\mathcal{C}\to \mathcal{Q}$ is locally surjective and hence an epimorphism of sheaves.

Cover the circle by the two arcs $U_0=S^1\setminus \{(-1,0)\}$ and $U_1=S^1\setminus \{(1,0)\}$. Choose continuous angle functions ${\theta }_0:U_0\to (-1\mathrm{/}2,1\mathrm{/}2)$ and ${\theta }_1:U_1\to (0,1)$ with $e^{2\pi i{\theta }_i(z)}=z$. Their quotient classes agree on $U_0\cap U_1$, because the difference ${\theta }_1-{\theta }_0$ is locally constant integer-valued there. Thus they glue to a section $q\in \mathrm{\Gamma }(S^1,\mathcal{Q})$. If $q$ came from a global continuous real function $g$, then $g-{\theta }_i$ would be an integer-valued continuous function on the connected set $U_i$, hence constant. On the two connected components of $U_0\cap U_1$ this would force the same constant difference to be both $0$ and $1$, which is impossible. So $\mathrm{\Gamma }(S^1,\mathcal{C})\to \mathrm{\Gamma }(S^1,\mathcal{Q})$ is not surjective.

Therefore an epimorphism of sheaves can fail to become surjective on global sections. This is exactly the phenomenon asserted in [F1], so the statement is refuted.