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Jun 21 at 19:08 comment added Mikhail Bondarko Yes, I understand that you get more exact functors if you vary the transformation in the definition. My question is whether it makes sense not to assume any functoriality for the corresponding isomorphisms at all.
Jun 21 at 16:16 comment added Fernando Muro @MikhailBondarko not if you equip it with the identity natural isomorphisms. The thing is that an exact functor must come equipped with such a natural transformation.
Jun 21 at 15:43 comment added Mikhail Bondarko Do you think that this identity is not exact in the sense of stacks.math.columbia.edu/tag/014V ?
Jun 21 at 14:47 history answered Dave Benson CC BY-SA 4.0