Sheafification of cohomology presheaf

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Given a complex of sheaves $F$ of abelian groups over a topological space $X$, I am trying to figure out whether the cohomology sheaf $H^n(F)$ is isomorphic to the sheafification of the presheaf $U\to H^n(\Gamma(U;F))$. I know that the two agree on stalks since the stalk functor is exact but is the one the sheafification of the other?