On Hartshorne Chapter II exercise 1.13

I am trying to solve the question about espace etale of a presheaf on Hartshrone (Chap II ex 1.13):

Given a presheaf $$\mathcal{F}$$, we define a topological space $$\mathrm{Spe(\mathcal{F})}=\coprod_{p\in X}\mathcal{F}_p$$ with the strongest topology such that for all the maps $$\bar{s}:U\rightarrow \mathrm{Spe(\mathcal{F})}$$ which sends $$P$$ to the germ $$s_P$$ for all $$U$$, and all $$s\in\mathcal{F}(U)$$ are continuous.

Show that for any $$U\subset X$$, the associated sheaf $$\mathcal{F}^+(U)$$ is the set of continuous sections of $$\mathrm{Spe(\mathcal{F})}$$ over $$U$$.

My try:

Here the only difficult part is to show any continuous section $$\bar{s}$$ of $$\mathrm{Spe(\mathcal{F})}$$ over $$U$$ is an element in $$\mathcal{F}^+(U)$$, i.e.

1. $$\bar{s}(P)\in \mathcal{F}_P$$ for any $$P\in U$$: this is clear from the definition of section;
2. for every $$P\in U$$, there exists a neighborhood $$V$$ and $$t\in \mathcal{F}(V)$$ such that for every $$Q\in V$$ we have $$s(Q)=t_Q$$.

I am struggling about the second part. Naturally, if we take a point $$P\in U$$, then $$s(P)=\langle V,t\rangle$$ for some open subset $$V\subset X$$ and $$t\in \mathcal{F}(U)$$. I don't know how to continue.

It seems that the only thing we can use here is $$\bar{s}$$ is continuous. So the expected neighborhood might be of the form $$\bar{s}^{-1}(W)$$ for some open set in $$\mathrm{Spe(\mathcal{F})}$$.

Any hints and answers are welcome!

Remark: Someone did some relevant work here, but I think there is a mistake at the third equivalence $$\cdots\iff \bar t(P) = \bar s (P) \iff \langle V,t\rangle_P=\langle U,s\rangle_P \iff \cdots$$ I think $$\bar{s}(P)=t_Q$$ cannot imply $$\langle V,t\rangle_P=\langle U,s\rangle_P$$ because we do not know what $$s$$ is.

However, if the above is true, then it will solve the problem. So probably I missed something obvious.

For every $$P\in U$$, we need to find a neighborhood $$V$$ and $$t\in \mathcal{F}(V)$$ such that for every $$Q\in V$$ we have $$s(Q)=t_Q$$.
Since $$\bar{s}(P)\in \mathcal{F}_P$$, so we can write $$\bar{s}(P)=\langle V, t\rangle$$ for some neighborhood $$V$$ of $$P$$ and $$t\in\mathcal{F}(V)$$. Then we have a "standard section" sending each point $$Q$$ to $$t_Q$$, the germ of $$t$$ at the point $$Q$$: $$\bar{t}:V\rightarrow \mathrm{Spe}(\mathcal F)$$
Consider the restriction of $$\bar{t}$$ on $$U\cap V$$, then $$\bar{t}(U\cap V)$$ is open since $$\bar{t}^{-1}(\bar{t}(U\cap V))=U\cap V$$ is open. Then take the neighbohood of $$P$$ as $$\bar{s}^{-1}(\bar{t}(U\cap V)).$$