Lemma 24.29.5. In the situation above, denote $RT : D(\mathcal{A}', \text{d}) \to D(\mathcal{B}, \text{d})$ the right derived extension of (24.29.4.1). Then we have

functorially in $\mathcal{M}$.

Lemma 24.29.5. In the situation above, denote $RT : D(\mathcal{A}', \text{d}) \to D(\mathcal{B}, \text{d})$ the right derived extension of (24.29.4.1). Then we have

\[ RT(\mathcal{M}) = Rf_* R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{N}, \mathcal{M}) \]

functorially in $\mathcal{M}$.

**Proof.**
By Lemmas 24.17.3 and 24.18.1 the functor (24.29.4.1) is right adjoint to the functor (24.28.0.1). By Derived Categories, Lemma 13.30.1 the functor $RT$ is right adjoint to the functor of Lemma 24.28.1 which is equal to $Lf^*(-) \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}$ by Lemma 24.28.3. By Lemmas 24.29.3 and 24.29.4 the functor $Lf^*(-) \otimes _\mathcal {A}^\mathbf {L} \mathcal{N}$ is left adjoint to $Rf_* R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (\mathcal{N}, -)$ Thus we conclude by uniqueness of adjoints.
$\square$

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