The problem is following.
1.13 Espace Etale of a Presheaf.
Given a presheaf $\mathscr F$ on $X$, we define a topological space $Spe(\mathscr F)$, called the espace etale of $\mathscr F$, as follows. As a set, $Spe(\mathscr F) = \bigcup _{P \in X} \mathscr F$. We define a projection map $\pi : Spe(\mathscr F )\to X$ by sending $s\in \mathscr F_P$ to $P$. For each open set $U \subseteq X$ and each section $s \in \mathscr F(U)$, we obtain a map $\bar s : U\to Spe(\mathscr F)$by sending $P \mapsto s_P$, its germ at $P$. This map has the property that $\pi \circ \bar s = id_U$, in other words, it is a "section" of $\pi$ over $U$. We now make $Spe(\mathscr F)$ into a topological space by giving it the strongest topology such that all the maps $\bar s : U\to Spe(\mathscr F)$ for all $U$, and all $s \in \mathscr F(U)$, are continuous. Now show that the sheaf $\mathscr F^+$ associated to $\mathscr F$ can be described as follows: for any open set $U \subseteq X, \mathscr F^+ (U)$ is the set of continuous sections of $Spe(\mathscr F)$ over $U$. In particular, the original presheaf $\mathscr F$ was a sheaf if and only if for each $U$, $\mathscr F (U)$ is equal to the set of all continuous sections of $Spe(\mathscr F)$ over $U$.
I understood all but the last line. If $\mathscr F$ was a sheaf, $\mathscr F$ is isomorphic to $\mathscr F^+$ and so for each $U$, $\mathscr F (U)$ is equal to the set of all continuous sections of $Spe(\mathscr F)$ over $U$. But the converse is unclear to me. In general the map $\theta : \mathscr F \to \mathscr F^+$ is not injective since the injectivity requires sheaf conditions. In this situation how can we regard the elements of $\mathscr F(U)$ as sections of $Spe(\mathscr F)$ over $U$? Is injective the map $\theta : \mathscr F \to \mathscr F^+$ even in the case of presheaf $\mathscr F$? $\theta$ is defined by $ s \mapsto (\bar s : P \mapsto s_P )$.
