Let $(\mathcal{F}_{\lambda})_{\lambda}$ a family of sheaves on the space $X$. Consider the sheaf product $\prod_{\lambda} \mathcal{F}_{\lambda} $.
My question is what are the weakest conditions for the $\mathcal{F}_{\lambda}$ such that the sheaf cohomology respects the product, therefore
$$ H^r(X, \prod_{\lambda} \mathcal{F}_{\lambda}) \cong \prod_{\lambda} H^r (X,\mathcal{F}_{\lambda})$$ holds (for $r>0$; case $r=0$ is always true because global section functor is right adjoint, so preserves limits) .
Remarks: This question arises from my former thread: Sheaf Cohomology of Injective Sheaves respects Products where the $\mathcal{F}_{\lambda}$ were injective.