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.

$\mathcal{E}xt^q(F, I)$ is 0 for $q > 0$. On the other hand it follows easily by considering extensions by 0 that $\mathcal{H}om(F, I)$ is flabby, hence acyclic. 

