Dear all, may be some of the readers of this list will know the answer to the following question. Let V be the category k-Mod for commutative ring k. For a finitely cocomplete V-category C, when is L = Lex[C^{op},V] abelian? I know some cases: 1. When C is abelian so is L. 2. When C is a free completion under finite colimits of a small category, L is abelian (because it's equivalent to a presheaf V-category). 3. L is reflective in the abelian [C^{op},V]. When the reflection is left exact L is abelian. However I don't any conditions that guaranty that the reflection is left exact. I would like to know some other conditions that ensure that L is abelian, and perhaps an example where L is not abelian. Thanks Ignacio [For admin and other information see: http://www.mta.ca/~cat-dist/ ]