sheaf theory - Describing global sections of sheafifications

The following is a counterexample for $mathcal{F}$ a presheaf of abelian groups (or sets, if you like).

Let $X=lbrace a,b,c,drbrace$ with nontrival opens given by $lbrace a rbrace,lbrace b rbrace,U=lbrace a,b,c rbrace,V=lbrace a,b,d rbrace, Ucap V$.

Define the presheaf $mathcal{F}$ by

$mathcal{F}(lbrace a rbrace)=mathcal{F}(lbrace b rbrace)=mathbb{Z}/2mathbb{Z}$,

$mathcal{F}(U)=mathcal{F}(V)=mathcal{F}(Ucap V)=mathcal{F}(X)=mathbb{Z}$,

with the obvious restriction maps.

Then $mathcal{F}^+(X)=lbrace (x,y)inmathbb{Z}oplusmathbb{Z}| xequiv ytext{ (mod 2)}rbrace$, since the germs at $a$ and $b$ are determined by those at $c$ and $d$, and the only restrictions on $c$ and $d$ are that they give the same germs at $a$ and $b$.

Consider $(0,2)inmathcal{F}^+(X)$. The germs at $c$ and $d$ cannot come from a common section of $mathcal{F}(X)$. Any system of sections which does not include $X$ in the cover must include both $U$ and $V$, having sections 0 and 2, respectively. Of course these do not agree when restricted to $Ucap V$. QED

This construction relies crucially on the fact that the presheaf is not separated (i.e. gluing is not unique). If the presheaf $it{is}$ separated, the condition described in the question is clearly satisfied.

This construction was shown to me by Paul Balmer.