when is Lex[A,V] abelian?

[Ignacio Lopez Franco <ill20@cam.ac.uk>]

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


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