Theories of presheaf type

Let us say that a geometric theory T is of presheaf type if its classifying topos B[T ] is (equivalent to) a presheaf topos. (We adhere to the convention that geometric logic allows arbitrary disjunctions, while coherent logic means geometric and finitary.) Write Mod(T ) for the category of Set-models and homomorphisms of T . The next proposition is well known; see, for example, MacLane–Moerdijk [13], pp. 381-386, and the textbook of Adámek– Rosický [1] for additional information: Proposition 0.1. For a category M, the following properties are equivalent: