How come $\{p,q\}$ maps to $\mathbb{Z}\oplus\mathbb{Z}$ in Wikipedia's example of a constant sheaf?
I expect $\{p,q\}$ to map to $\mathbb{Z}\otimes\mathbb{Z}$, in particular I'd define the global element as the tuple $\langle m\in F\{p\},n\in F\{q\}\rangle$ in order to comply with the gluing axiom. Also, the article mentions two projections $\pi_1, \pi_2$ to $\mathbb{Z}$. This sounds very wrong: the categorical coproduct does not have projections but inclusion arrows going in the other direction.
Can someone explain?
I do find that Wikipedia page to be quite poorly written.
The sections of this constant sheaf over $\{p,q\}$ are the locally constant maps from $\{p,q\}$ to $\Bbb Z$. In both cases the spaces have the discrete topology. Every map $X\to Y$ where $X$ and $Y$ have the discrete topology is both continuous and locally constant. (Recall $f:X\to Y$ is locally constant if all its fibres $f^{-1}(y)$ are open in $X$). Thus the sections are all maps from $\{p,q\}$ to $\Bbb Z$. These form the group $\Bbb Z^2$.
As you say sections have the form $\langle m,n\rangle$ for $m\in F\{p\}$, $n\in F\{q\}$ but as both of these groups are $\Bbb Z$, the sections are integer pairs $\langle m,n\rangle$.
Also in an Abelian category, for instance the category of Abelian groups, $A\oplus B$ is a product as well as a coproduct. So there are inclusions and projections. Some texts call such a beast a "biproduct".