About your last sentence, that surjections are not necessarily quotients, are there a subset of geometric surjections that do just that? An intermediate step before full surjections and hopefully a three-way factorization system?

So, you are saying, since there is no device like a subobject classfier that classifies quotients, we cannot hope for an analogous result.

You're right. I should be asking about surjections out of $\mathcal{E}$. Even an ionad is a surjection out of $Set^X$.

Explicitly, is the classifying map $j: \Omega \to \Omega$ of $T \subseteq \Omega$ the same as the local modality defining the sheaf topos? I guess this is the meaning of the phrase 'classifier of dense sub-presheaves'.

Excellent answer, thank you! Is there any relationship between $T \subseteq \Omega$ and the subobject classifier of the topos of sheaves for $T$?

What if the category has all pullbacks? Then I guess the base-change functor will do, won't it?

Thanks Buschi, that is helpful but additionally I want to know what kind of diagrams they correspond to. Is there any relation with the diagram being flat as a functor $J^{op} \to C^{op}$ or something like that?

Now I am studying your paper and its Twelf companion. It seems interesting. Thanks for the link.