The category of internal locales in the Grothendieck topos of sheaves on a locale X is equivalent to the slice category over X. In other words, internal locales over X are precisely morphisms of locales of the form Y→X.

**Is the same statement true for smooth manifolds instead of locales
and submersions instead of continuous maps?**

More precisely, consider the site of smooth manifolds equipped with its standard Grothendieck topology of surjective submersions.

**Is the category of internal smooth manifolds in the Grothendieck topos of sheaves
on a smooth manifold M equivalent to the category whose objects are surjective
submersions of the form N→M and morphisms are commuting triangles?**

Definitions tend to branch when we internalize them, so an answer to this question should include the correct definition of a smooth manifold in an arbitrary Grothendieck topos.

According to nLab, internal dualizable modules over the locale of real numbers in the category of sheaves on a smooth manifold M are precisely finite-dimensional vector bundles over M. This can be seen as a further motivation for the above question.