I know that the sheafification functor $a:Pr(C) \to Sh(C,J)$ is up to equivalence the localisation $Pr(C) \to Pr(C)[W^{-1}]$ at the class of those morphisms which $a$ inverts. But is it also the localisation at the class of covering sieve inclusions $S \to y(c)$?
I tried to show that any functor $g:Pr(C) \to E$ which inverts all covering sieve inclusions mustalso invert all morphism which $a$ inverts, but I wasn't successful. It works when $g$ preserves finite limits and colimits though.
I suspect that it is false, but I would love to see a counterexample!
Let's make things as simple as possible. Consider $\mathcal{C} = \{ a \to 1 \}$ and let $J$ be the Grothendieck topology generated by $a \to 1$. The category of $J$-sheaves is equivalent to $\textbf{Set}$. Specifically, we can identify the inclusion $\textbf{Sh} (\mathcal{C}, J) \hookrightarrow \textbf{Psh} (\mathcal{C})$ with $\Delta : \textbf{Set} \to \textbf{Psh} (\mathcal{C})$. The left adjoint can be identified with evaluation at $a$.
On the other hand, the only $J$-covering sieve (other than the trivial ones) is the one generated by $a \to 1$. Consider the functor $E : \textbf{Set}^\textrm{op} \to \textbf{Set}$ sending a set $X$ to the set $E (X)$ of partitionings of $X$ (or, equivalently, the set of equivalence relations on $X$). This inverts $\emptyset \to 1$: after all, they both have exactly one partitioning. Therefore the functor $\textbf{Psh} (\mathcal{C}) \to \textbf{Set}^\textrm{op}$ sending $F$ to $E (F (1))$ inverts $h_a \to h_1$. But it does not invert $h_a \amalg h_a \to h_1 \amalg h_1$. Thus inverting the $J$-covering sieves (as subobjects of representables) in $\textbf{Psh} (\mathcal{C})$ is different from inverting all $J$-local isomorphisms.