For a category, write
for its presheaf topos.
For any topos, write
There is an equivalence of categories
This equivalence takes to the composite
See for instance (Johnstone, theorem B3.2.7).
If is a finitely complete category we may think of it as the syntactic category and in fact the syntactic site of an essentially algebraic theory . An internally flat functor is then precisely a finite limit preserving functor, hence is precisely a -model in .
Therefore the above theorem says in this case that there is an equivalence of categories
between the geometric morphisms and the -models in .
This says that is the classifying topos for .
In this case an internally flat functor may be identified with a -torsor object in .
For this reason one sees in the literature sometimes the term “torsor” for internally flat functors out of any category . It is however not so clear in which sense this terminology is helpful in cases where is not a delooping groupoid or at least some groupoid.
A standard textbook references is section B3.2 in
The first proof of this result can be found in:
Another proof is in