I was reading this article http://ncatlab.org/nlab/files/cech.pdf and I could not understand the construction of the left adjoint of the inclusion $\mathbf{PSh}(C, J) \rightarrowtail \mathbf{Sh}(C, J)$ in page 14. Usually the condition for $F$ being a sheaf is that there exists a unique lifting of $R \rightarrow F$ to $h_U$ for every $U$ and $ R \in J(U)$, however the author uses a coequalizer (in the category $\mathbf{PSh}(C, J)$) instead a of sieve, and I cannot see the equivalence between these two definitions (I know that $Nat(R, F)$ corresponds to a compatible family of elements).
Furthermore, I cannot understand the functor $\mathscr{B}$ created as a pushout? What would be $\mathscr{B}$ in a topological space instead of a site?
Moreover, the construction iterating the applications of $\mathscr{B}$ and $\mathscr{A}$ seens totally unintuitive and I have no idea how to prove that this colimit is in fact left adjoint to the inclusion. The unique construction that I have ever saw of the sheaffication in sites is applying ${+}$ two times, where $F^{+}(c) = colim_{R \in J(c)} R$.
Thanks in advance.