Hi,
I'm currently reading through "Sheaves in Geometry and logic" by Mclane-Moerdijk and this one issue has been bugging me for a long time, which I hope you could help me resolve. 


It is known that for a general presheaf on a Grothendieck Topology, we must in general apply the plus construction twice to obtain a sheaf. The first application turns an arbitrary presheaf into a separated presheaf, and one more application gives a sheaf. So, exactly, intuitively, what obstructs us from getting a sheaf from just one application of the plus construction when the sheaf is non-separated? Let us take the example:
[What is an example of a presheaf P where P^+ is not a sheaf, only a separated presheaf?](https://mathoverflow.net/questions/616/what-is-an-example-of-a-presheaf-p-where-p-is-not-a-sheaf-only-a-separated-pre) 
given by Sherry. When we apply it once we get a separated presheaf, OK. But what exact component of that presheaf hindered us from getting the sheaf we wanted? I agree with what Sherry wrote in that case, namely that : "So in our example, 1 and 3, over ABC and BCD, in our original presheaf were compatible on a refinement of BC but not on BC"
Can we generalize this notion to become rigorous in the case for arbitrary non-separated presheaves?



Sheafification - Why does twice suffice?
If I understand you correctly you believe in the counter example and just want an intuitive reason. Here is how I think about it (I am open for remarks!):


Definition:
Let $\mathscr A$ be a (small) category with arbitrary (small) limits. A presheaf $F$ on a (small) site $\mathscr C$ with values in $\mathscr A$ is a sheaf, if the diagrams
$F(U)\rightarrow \prod\_i F(U\_i)\rightrightarrows \prod\_{i,j}F(U\_i\times\_UU\_j)$
are equalizers for all covering families $U\_i\rightarrow U$ of all objects $U$ of $\mathscr C$.


Construction:
Let $F$ be an $\mathscr A$-valued presheaf on a site $\mathscr C$ and $U\in \mathscr C$. Then for any covering family $\mathscr U=\{U\_i\rightarrow U\}$, define $F\_U(\mathscr U)$ to be the equalizer of the sequence
$F\_U(\mathscr U)\rightarrow \prod\_i F(U\_i)\rightrightarrows \prod\_{i,j}F(U\_i\times\_UU\_j).$
Further define $F^+(U)$ as the colimit
$F^+(U)=colim\_{\mathscr U}F\_U(\mathscr U)$.


And here is what happens: We do the whole construction for each $U$ seperately. And this is precisely the dangerous part: Look at the set $U=ABC$ in Sherry's example. Here, the set $BC$ is part of a covering and thus has influence on $F^+(U)$. But at the same time we apply our construction to the set $BC$, which doesn't even see the element $A$. So while we are trying to adjust our sheaf on $ABC$, so that it fits with $BC$, we change the sheaf on $BC$ independently. In the end more sections are compatible than before, but we didn't include their glueings yet.
Morally the reason for that phenomenon is (non-)separatedness. By Definition we just have precisely one glueing for every compatible family in $F^+$. That means on the one hand that $F^+$ is automatically separated, but it means also that if $F$ is not separated then the different glueings for the same compatible family become "identified". In our example we have the elements $1$ and $3$ in $F(BC)$, which give the same compatible family in $F(C)$ and $F(B)$. In $F^+(BC)$ we have just *one* element for every family in $F(C)$ and $F(B)$.So $F^+(BC)$ looks quite different than $F(BC)$.


