Global sections are left exact but need not preserve epimorphisms

Suppose $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ is exact. By [L1], the kernel of the map $\mathrm{\Gamma }(X,\mathcal{F})\to \mathrm{\Gamma }(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ is the section group of the kernel sheaf at $X$. By [L2], that kernel sheaf is the image of ${\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}$, so its sections over $X$ are exactly the image of $\mathrm{\Gamma }(X,{\mathcal{F}}^{\mathrm{\prime }})\to \mathrm{\Gamma }(X,\mathcal{F})$. Together with [F1], this is the displayed left exactness.

For the failure of surjectivity, let $X=S^1$, 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 quotient sheaf obtained by sheafifying the presheaf $V\mapsto \mathcal{C}(V)\mathrm{/}\underset{‾}{\mathbb{Z}}(V)$. Every germ of a section of $\mathcal{Q}$ is represented locally by a continuous real-valued function, so the canonical map $\mathcal{C}\to \mathcal{Q}$ is locally surjective and hence an epimorphism of sheaves.

Cover the circle by the 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 $e^{2\pi i{\theta }_i(z)}=z$. On one connected component of $U_0\cap U_1$ the difference ${\theta }_1-{\theta }_0$ is $0$, and on the other it is $1$, so the classes of ${\theta }_0$ and ${\theta }_1$ agree in the quotient presheaf and glue to a global section $q\in \mathrm{\Gamma }(X,\mathcal{Q})$. If $q$ had a lift $g\in \mathrm{\Gamma }(X,\mathcal{C})$, then $g-{\theta }_i$ would be an integer-valued continuous function on the connected arc $U_i$, hence a constant $n_i$. On the two components of $U_0\cap U_1$ this would force $n_0-n_1$ to be both $0$ and $1$, a contradiction. Thus $\mathrm{\Gamma }(X,\mathcal{C})\to \mathrm{\Gamma }(X,\mathcal{Q})$ is not surjective.

Step 1.1 gives left exactness, and step 1.3 gives an epimorphism whose global sections map is not surjective.