vanishing cohomology for indiscrete Grothendieck topology

Question

Let $\mathcal{C}$ be a category endowed with the indiscrete Grothendieck topology. Let $X$ be an object of $\mathcal{C}$ and $\mathcal{F}$ be an abelian sheaf on $\mathcal{C}$. Then I want to know how to prove that $H^q(X, \mathcal{F})=0$ for all $q>0$.

This fact is used in the proof of this lemma.

Answer 1

In the indiscrete topology, sheaves are the same as presheaves. So, the cohomology functors $H^q(X,-)$ are the derived functors of the functor $\mathcal{F}\mapsto\mathcal{F}(U)$ on presheaves. This functor is exact, since limits and colimits of presheaves are computed pointwise.