Suppose $\mathbb{T}$ is a geometric theory, $\mathcal{E}$ is a topos, and $M$ is a model of $\mathbb{T}$ in $\mathcal{E}$. Is there any sort of elementary condition on $M$ and $\mathcal{E}$ (or, even better, on the geometric morphism $\mathcal{E}\to \mathbf{Set}$) which would allow us to recognize $\mathcal{E}$ as the classifying topos of $\mathbb{T}$ and $M$ as the generic $\mathbb{T}$-model therein?
I feel like this is a long shot, but I thought I would ask anyway.
Edit: Of course, such a condition could not be expressed in the internal logic of $\mathcal{E}$ (even including non-geometric logic), since then it would be preserved in all slices $\mathcal{E}/X$. This is one reason I feel it's a long shot; but the example of principal bundles mentioned in the comments suggests that it's not an entirely unreasonable question.
