Let $F$ be a presheaf on a topological space $X$ of some category of "sets with structure." In Borel's Linear Algebraic Groups, he gives the following explanation for how to construct the associated sheaf $F'$:
Roughly speaking, $F'$ can be constructed in two steps. First, define $F_1(U)$ to be $F(U)$ modulo the equivalence relation which relates $s$ and $t$ if their restrictions agree on some open cover of $U$. Then form $F'(U)$ by "adding" to $F_1(U)$ all elements obtainable from compatible local data on some open covering of $U$. This process makes sense thanks to step 1.
Unfortunately, this process does not make sense. I permit no thanks to step 1. Anyway, this differs from other constructions of sheafification which I have seen before. The main one I am familiar with is to define $F'(U)$ to be the set of functions $f$ from $U$ into the disjoint union of stalks $F_x : x \in U$ such that the following holds: for each $x \in U$, $f(x) \in F_x$, and there is an open covering $U_i$ of $U$, and sections $s_i \in F(U_i)$, such that $f(x) = (s_i,U_i)_x$ for each $x \in U_i$.
Is there a nice way to understand what Borel is talking about? I just don't get it. Is there a nice way to relate these two notions of sheafification together? (besides universal property shenanigans)