Sheafification of a presheaf through the etale space

Question

I have some problems to show that the following contruction defines a sheafification:

Let $\mathcal F$ be a presheaf on $X$, and let $Et(\mathcal F)$ be the etale space associated to $\mathcal F$, with $π:Et(\mathcal F)\rightarrow X$ that is the canonical map which sends a germ $s_x$ in $x$. If with $U$ we indicate a generic open set of $X$, then the set of sections of $π$ on $U$ is

$\mathcal {F}^+(U)=$ {$\widetilde{s}:U\rightarrow Et(\mathcal F)\;with\;\widetilde {s}(x)=s_x\; \forall s\in\mathcal {F}(U)$}

We give a certain topology on $Et(\mathcal F)$ and make $π$ and $\widetilde s$ continuous functions. In this way whe define the sheaf $\mathcal F^+$ of continuous sections of $π$, and the morphism (for all $U$)

$\phi(U):\mathcal F(U)\rightarrow\mathcal F^+(U)$ such that $s\mapsto\widetilde s$

Now if $\mathcal F^+$ satisfies the "universal property", it is the sheafification of $\mathcal F$. Suppose that $\psi$ is a morphism from $\mathcal F$ in a generic sheaf $\mathcal G$; how can I prove that exists a unique morphism $\theta:\mathcal F\rightarrow\mathcal G$ such that $\psi=\theta\circ\phi$?

The definition $\theta(\widetilde s)=\psi(s)$ doesn't work because $s_x=t_x$ for all $x$ doesn't imply $s=t$ in $\mathcal F(U)$ since $\mathcal F$ is a presheaf.

Answer 1

Given some section $\tilde{s} : U \to \mathrm{Et}(\mathcal{F})$, we have for all $x \in U$ some element in $\mathcal{F}_x$ and therefore in $\mathcal{G}_x$. The continuity of the map $\tilde{s}$ ensures that we can actually lift these germs to local sections around $x$. Since $\mathcal{F} \to \mathcal{G}$ is a homomorphism of presheaves, it doesn't matter on which neighborhood we work, and since $\mathcal{G}$ is a sheaf, we can glue these local sections to some section in $\mathcal{G}(U)$. The rest is also easy, you should be able to do this on your own.

Answer 2

The construction of the sheafification you mention is nicely described e.g. in Tennison's book Sheaf Theory, in all details . The functor $\Gamma$ from the category of sheaf spaces to the category of presheaves, giving the sheaf of sections of a sheaf space, and the functor $L$ (your Et), from the category of presheaves to the category of sheaf spaces, are a pair of adjoint functors, and the composition $\Gamma L$ is the wanted reflection.