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Let $(R, \mathfrak m, k)$ be a commutative Noetherian local ring. Then, is it true that $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commutes with arbitrary co-products?

I'm trying to build a little dictionary between old Homological algebra for local rings and the slightly more modern approach via derived functors. Let $X = SpecA$ be a spectrum of a local ring $(A,...