10 ANDREJ gfiEDROV

the category Mod(F,S ) and the category of F-internal flat functors on

C (as a constant object in F) and F-internal natural transformations.

Moreover, this equivalence is natural in F •

Remark. A few comments on the statement of the theorem are in order.

If F is the category of S , it just says that "the category

Mod(F,SCCop)

C C

is equivalent to the full subcategory of S whose objects are flat

functors on CC" . The statement of the theorem means that the above

sentence in quotation marks is true when interpreted in F . In practice,

it often suffices to look at the case f - S , and check that the

facts about the situation are proved intuitionistically.

Corollary (Diaconescu). Under the hypotheses of the theorem, let J be

a Grothendieck topology on C C . Then there is an equivalence between the

category Mod(F,Shj (CC ) ) and the category of F-internal flat continuous

functors on C C (i.e. F-internal flat functors on ( C taking J-covers to

epimorphic families in F).

In practice, J is the Grothendieck topology generated by a family

of sieves on C C closed w.r.t. pullbacks:

Lemma (folklore) . Let K be a family of sieves in a category C C , and

assume that K is closed w.r.t. pullbacks. Then for the Grothendieck

topology J generated by K , a flat functor from C C to a Grothendieck

topos F is continuous for J iff it sends every sieve in K to an

epimorphic family in F .