# Finite direct sum of injective resolutions

I have two sheaves of modules $\mathcal F$ and $\mathcal G$ over some scheme $X$. What I want to do is really to compare the sheaf cohomologies of $\mathcal F, \mathcal G$ and $\mathcal F \oplus \mathcal G$. Given an injective resolution $I^*_\mathcal F$ of $\mathcal F$ and $I^*_\mathcal G$ of $\mathcal G$, $I^*_\mathcal F \oplus I^*_\mathcal G$ is a coresolution of $\mathcal F \oplus \mathcal G$, but is it injective?

I also assume that direct sums over sheaves commute with sections, so that $\Gamma(-, \mathcal F \oplus \mathcal G)$ is isomorphic to $\Gamma(-, \mathcal F) \oplus \Gamma(-, \mathcal G)$, and that $\mathcal F \oplus \mathcal G$ is actually a sheaf.

Also by general abstract nonsense, there is a canonical isomorphism $\Gamma(-, A \times B \times C \times \cdots) \cong \Gamma(-, A) \times \Gamma(-, B) \times \Gamma(-, C) \times \cdots$ because $\Gamma(Y, -)$ is a representable functor for any open set $Y$... at least in the category of sheaves of sets, the category of abelian sheaves, and the category of $\mathscr{O}_X$-modules.