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This a paragraph from Hartshorne's Algebraic Geometry:

Next we will define a sheaf of rings $\mathscr{O}$ on $\text{Spec }A$. For each prime ideal $\mathfrak{p}\subseteq A$, let $A_{\mathfrak{p}}$ be the localization of $A$ at $\mathfrak{p}$. For an open set $U\subseteq \text{Spec }A$, we define $\mathscr{O}(U)$ to be the set of functions $s:U\to \coprod_{\mathfrak{p}\in U}A_{\mathfrak{p}}$, such that $s(\mathfrak{p})\in A_{\mathfrak{p}}$ for each $\mathfrak{p}$, and such that $s$ is locally a quotient of elements of $A$: to be precise, we require that for each $\mathfrak{p}\in U$, there is a neighborhood $V$ of $\mathfrak{p}$, contained in $U$, and elements $a, f\in A$, such that for each $\mathfrak{q}\in V$, $f\notin \mathfrak{q}$, and $s(\mathfrak{q})=a/f$ in $A_{\mathfrak{q}}$. (Note the the similarity with the definition of the regular functions on a variety. The difference is that we consider functions into the various local rings, instead of to a field.)

My Question: $\mathscr{O}$ seems to be the sheafification of a presheaf. Is it? What is it?

I have compared the construction of the sheafification of a presheaf:

We construct the sheaf $\mathscr{F}^+$ as follows. For any open set $U$, let $\mathscr{F}^+(U)$ be the set of functions $s$ from $U$ to the union $\bigcup_{P\in U}\mathscr{F}_P$ of the stalks of $\mathscr{F}$ over points of $U$, such that

(1) for each $P\in U$, $s(P)\in \mathscr{F}_P$, and

(2) for each $P\in U$, there is a neighborhood $V$ of $P$, contained in $U$, and an element $t\in \mathscr{F}(V)$, such that for all $Q\in V$, the germ $t_Q$ of $t$ at $Q$ is equal to $s(Q)$.

I think it should have $$\varinjlim_{\mathfrak{p}\in U}\mathscr{F}(U)=\mathscr{F}_{\mathfrak{p}}=A_{\mathfrak{p}}=\varinjlim_{f\in A\backslash \mathfrak{p}}A_f. $$ But I don't know how to defin $\mathscr{F}(U)$...

Edit. I know that every sheaf is a presheaf itself. However, we have not proven that $\mathscr{O}$ is a sheaf. It is the task of the following paragraph in the book.

Now it is clear that sums and products of such funtions are again such, and that the element $1$ which gives $1$ in each $A_{\mathfrak{p}}$ is an identity. Thus $\mathscr{O}(U)$ is a commutative ring with identity. If $V\subseteq U$ are two open sets, the natural restriction map $\mathscr{O}(U)\to \mathscr{O}(V)$ is a homomorphism of rings. It is then clear that $\mathscr{O}$ is a presheaf. Finally, it is clear from the local nature of the definition that $\mathscr{O}$ is a sheaf.

That is why I am looking for a presheaf which induces $\mathscr{O}$.

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