Quasi-closed local operator ($q(U)(p)=((p\Rightarrow u)\Rightarrow u)$)

For a subterminal truth value $u$, the quasi-closed local operator is relative double negation $q(U)(p)=((p\Rightarrow u)\Rightarrow u)$. It has bottom value $q(U)(0)=u$. Its fixed truth values are Boolean with relative bottom $u$, inherited meet, join closed by $q(U)$ and complement $p\Rightarrow u$. Therefore its sheaf topos is a Boolean topos.

In $\mathcal{E}/\Omega$, the generic truth mono $\top :1\hookrightarrow \Omega$ defines a quasi-closed Boolean subtopos. Its composite to $\mathcal{E}$ is surjective. If a predicate $p(x)$ pulled back from $\mathcal{E}$ becomes dense, then $((p(x)\Rightarrow u)\Rightarrow u)=1$ for the generic truth $u$. Substitution $u=p(x)$ yields $p(x)=1$, proving reflection of invertible monos and hence faithfulness.