Let $(\rho_{\lambda})_{\lambda}$ a family of injective sheaves on the space $X$ (therefore injective objects on the category of abelian sheaves). Using universal property of product it's easy to see that $\prod_{\lambda} \rho_{\lambda} $ is also an injective sheaf.
My question is why then for the sheaf cohomology holds the isomorphism
$$ H^r(X, \prod_{\lambda} \rho_{\lambda}) \cong \prod_{\lambda} H^r (X,\rho_{\lambda})$$
Ideas:
I know that the sheaf cohomogies induced by injective and acyclical resolutions are identical, but I don't see why this fact may inply the isomorphism above.
Since injective sheaves are acyclic for global sections, the statement is trivial for $r > 0.$ So all you're claiming is that global sections of the product is the product of global sections. There are various ways to see this -- for example, you could check that the presheaf (sectionwise) product is actually a sheaf, or you could observe that global sections is representable.