Lemma 47.13.7. In Situation 47.13.6 the functor R\mathop{\mathrm{Hom}}\nolimits (A, -) is equal to the composition of R\mathop{\mathrm{Hom}}\nolimits (E, -) : D(R) \to D(E, \text{d}) and the equivalence - \otimes ^\mathbf {L}_ E A : D(E, \text{d}) \to D(A).
Proof. This is true because R\mathop{\mathrm{Hom}}\nolimits (E, -) is the right adjoint to - \otimes ^\mathbf {L}_ R E, see Differential Graded Algebra, Lemma 22.33.5. Hence this functor plays the same role as the functor R\mathop{\mathrm{Hom}}\nolimits (A, -) for the map R \to A (Lemma 47.13.1), whence these functors must correspond via the equivalence - \otimes ^\mathbf {L}_ E A : D(E, \text{d}) \to D(A). \square
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