Let $\mathcal{R}_X$ be a sheaf of (commutative) rings on a topological space $X$. Let $\mathcal{M}_X, \mathcal{N}_X, \mathcal{P}_X$ be $\mathcal{R}_X$-modules.
Do the derived $\mathrm{Hom}$-functors commute with (finite) products? Meaning, do we have an isomorphism in the derived category of $\mathcal{R}_X$-modules:
$$\mathrm{Hom}_{\mathbf{D}(\mathcal{R}_X)}(\mathcal{M}_X\oplus \mathcal{N}_X, \mathcal{P})\cong \mathrm{Hom}_{\mathbf{D}(\mathcal{R}_X)}(\mathcal{M}_X, \mathcal{P})\oplus \mathrm{Hom}_{\mathbf{D}(\mathcal{R}_X)}(\mathcal{N}_X, \mathcal{P}_X)?$$
As far as I know the following isomorphism
$$\mathrm{Hom}_{\mathbf{D}(\mathcal{R}_X)}(\mathcal{M}_X, \mathcal{N}_X\oplus\mathcal{P})\cong \mathrm{Hom}_{\mathbf{D}(\mathcal{R}_X)}(\mathcal{M}_X, \mathcal{N}_X)\oplus \mathrm{Hom}_{\mathbf{D}(\mathcal{R}_X)}(\mathcal{M}_X,\mathcal{P})$$
actually does exist. (see for example Lemma 33.9 in https://stacks.math.columbia.edu/download/derived.pdf). But I cannot find an explanation when the direct sum is in the first argument of the derived $\mathrm{Hom}$ functor.