Explicit sections after sheafification Hartshorne (in his book on algebraic geometry) constructs, for a presheaf $\mathscr{F}$ over a topological space $X$, the sheafification of $\mathscr{F}$, $\mathscr{F}^+$, as
$$ \mathscr{F}^+(U) = \{ s\!: U\to \bigsqcup_{p\in U}\mathscr{F}_p \text{ such that }(*) \} $$
with $(*)$ being the following conditions:


*

*for each $p\in U$, $s(p)\in\mathscr{F}_p$.

*for each $p\in U$, there is a neighbourhood $V\subset U$ of $p$, and some $t\in\mathscr{F}(V)$, such that for any $q\in V$, the germ $t_q = s(q)$.


My question essentially boils down to the following (and I'll expand on this):

What, explicitly, are the elements of $\mathscr{F}^+(U)$?

What I mean is that, while the description above is in some ways explicit, it leaves too many details out (at least for me), and so I want some "classification" of the various kinds of sections we can get.
For example, it is clear that any element $s\in\mathscr{F}^+(U)$ is of the form $s\!: p\mapsto t_p$, with $t$ not necessarily being fixed. The case where it is fixed, of course, is the image of the natural morphism $\theta\!:\mathscr{F}\to\mathscr{F}^+$, since this is given by $\theta(s)\mapsto (p\mapsto s_p)$. This gives some of the elements of $\mathscr{F}^+(U)$, but not all of them.
As far as I've been able to tell, any section $a\in\mathscr{F}^+(U)$ will be determined by considering an open cover $\{U_i\}_{i\in I}$ of $U$, with classes of sections $[t^i]\subset\mathscr{F}(U_i)$, such that all $s\in[t^i]$ agree on germs (i.e. for any $s,v\in[t^i]$ and $p\in U_i$ we have $s_p = v_p$) and, similarly, for any $s\in[t^i]$, $v\in[t^j]$, and $p\in U_i\cap U_j$ we have $s_p = v_p$ (that is, the germs agree on intersections).
So, a secondary question is:

Is the above characterization of the sections of $\mathscr{F}^+$ correct?

 A: So, basically the sections of the associated sheaf are functions which are locally sections of the presheaf. Maybe a clear way to see this is the following example: take a smooth manifold $M$ and define a presheaf $\mathscr{F}$ by $\mathscr{F}(U)=\{f:U\to \mathbb{R}:f\:\text{is constant}\}$. If $M$ is not connected, then write $M_1\sqcup M_2$ for instance, both clopen. We can write a function $f_1:M_1\to \mathbb{R}$ by $f_1(x)=1$ and $f_2:M_2\to \mathbb{R}$ by $f_2(x)=2$. Because $M_1\cap M_2=\varnothing$, if $\mathscr{F}$ were a sheaf, then the piecewise function $f:M\to \mathbb{R}$ given by
$$f(x)= \begin{cases}
f_1(x)&x\in M_1\\
f_2(x)& x\in M_2.
\end{cases}$$
would have to be in the sheaf. However, it is not because $f$ is not constant globally, only locally. Once we pass to the sheafification, $\mathscr{F}^+$, our sections on $U\subseteq M$ are 
$$ \mathscr{F}^+(U)=\{f:U\to \mathbb{R}:(1)\:\&\:(2)\}.$$
$(1)$ is the property that $f(P)\in \mathscr{F}_P$. $\mathscr{F}_P=\mathbb{R}$ so this is fine.
$(2)$ is the property that around each $P\in U$, there exists a neighborhood $V\subseteq U$ with $f|_V(P)=s_P$ for $s\in \mathscr{F}(U)$. That is, $f$ is locally a member of $\mathscr{F}$. That is, $f$ is locally constant.
So, the sections of $\mathscr{F}^+(U)$ are locally constant functions valued in $\mathbb{R}$. Morally, a sheaf is supposed to be an object that locally assigns functions on the space, and the sheafification formalism says that indeed this is the case. 
