Is the subobject classifier of the sheaves the sheaffication of the one from the presheaves?

Question

maybe this is an idiot question, but I could not solve it. Let $\Omega$ be the suboject classifier in the category $\mathbf{PSh}(X, J)$ where $(X, J)$ is a site, I know that $\Omega(U) \cong Nat(h_U, \Omega) \cong $subpresheaves of $h_U$. Now let $\tilde\Omega$ be the subobject classifier in $\mathbf{Sh}(X, J)$. Is $\Omega^{++} \cong \tilde\Omega$ ?

Answer 1

No. Consider presheaves on $\mathbb{2} = \{ 0 \to 1 \}$. We may equip $\mathbb{2}$ with the Grothendieck topology generated by the empty sieve on $0$. Then sheaves on this site are those presheaves $F : \mathbb{2}^\mathrm{op} \to \mathbf{Set}$ such that $F (0)$ is a singleton. In particular, the category of sheaves is equivalent to $\mathbf{Set}$. The subpresheaf classifier has $\Omega (0) = \{ \emptyset, \{ 0 \to 0 \} \}$ and $\Omega (1) = \{ \emptyset, \{ 0 \to 1 \}, \{ 0 \to 1, 1 \to 1 \} \}$, so its associated sheaf, regarded as a set, has three elements. But the subobject classifier of $\mathbf{Set}$ has two elements.

The correct construction is as follows: given a site $(\mathcal{C}, J)$ , the subobject classifier for $\mathbf{Sh}(\mathcal{C}, J)$ is the sheaf $\tilde{\Omega}$ where $\tilde{\Omega} (C)$ is the set of subsheaves of the sheaf associated with the representable presheaf on $C$.