In Algebraic Geometry Ⅱ by Mumford and Oda, page 294, some reasonable cases where the long exact cohomology sequences for sheaves holds are being analyzed.
For example, if we could prove
($*$)for all presheaves $\mathcal{F}$, the canonical maps $H^i(X,\mathcal{F}) \to H^i(X,sh(\mathcal{F}))$ are isomorphisms.
then the long exact sequences holds.
Then the authors said
Maybe applying twice the long exact sequence means applying to the two short exact sequences in the diagram to show the morphisms induced by $\mathcal{F} \to \mathcal{F}'$ and $\mathcal{F}' \to sh \mathcal{F}$ are isomorphism. But I can't see why we ($**$) could imply ($*$). Could you give me some help? Thanks!