Why is $H^p(X, \mathcal{E}xt^q(F, I))=0$ for $p>0$ and $I$ an injective sheaf and $F$ an arbitrary sheaf? I'm trying to check the hypothesis in the grothendieck spectral sequence applied to the functor $\mathcal{E}xt(F, -)$ and global sections functor, and so I'm trying to see why the sheaf $\mathcal{E}xt(F, I)$ is acylic for the the global sections functor, which I think translates to my question in the first sentence.