I believe the following theorem is correct. I would really like it to be, at least for Grothendieck toposes, and it may be well known. But I want to make sure I have not missed something. The motivation is Barr covers but I believe it is a result in elementary topos theory. Theorem: For any topos A and geometric morphism f^*,f_*:B-->A, where B satisfies axiom of choice, the direct image f_* preserves module quotients. Proof: In the choice topos B every quotient homomorphism has a right inverse function (which is generally not a module homomorphism), so its direct image also does, so the direct image is onto and thus is a quotient. Is that good? Colin [For admin and other information see: http://www.mta.ca/~cat-dist/ ]