It is obvious that there is a parallel between the definition of structure sheaf of spec(A) versus the sheafification of a pre-sheave.

The definition of the sheave $\mathscr F^+$ associated to pre-sheave $\mathscr F$ is (Hartshorne p.64): For any open set $U$, let $\mathscr F^+ (U)$ be the set of function $s$ from $U$ to the union of stack $\mathscr F_P$ of $\mathscr F$ over points $P$ of $U$ such that: (1) For each $P$ in $U$, $s(P)$ is in $\mathscr F_p$. (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 $s(Q)$.

While, in (Hartshorne p.70), the definition of the sheaf of ring $\mathscr O$ on $Spec(A)$ is: For any open set $U$ of $Spec(A)$, let $\mathscr O(U)$ be the set of function $s$ from $U$ to the union of localization $A_\mathscr{p}$ of $A$ at $\mathscr{p}$ such that: For each $\mathscr{p}$ in $U$, there is a neighborhood $V$ of $\mathscr{p}$ , contained in $U$, and elements $a,f$ of $A$, such that for each $\mathscr{q}$ in $V$, with $f$ not in $\mathscr{q}$, we have $s(\mathscr{q)}) = a/f$ .

So, is there a pre-sheave on $Spec(A)$ (which in general is not a sheave) that exits for any ring $A$ such that its sheafification gives exactly the structure sheaf $\mathscr O$ of $spec(A)$?