Proposition 48.32.2. In Situation 48.16.1 let $f : X \to Y$ be a morphism of $\textit{FTS}_ S$. Then the functors $Rf_!$ and $f^!$ are adjoint in the following sense: for all $K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ and $L \in D^+_{\textit{Coh}}(\mathcal{O}_ Y)$ we have

\[ \mathop{\mathrm{Hom}}\nolimits _ X(K, f^!L) = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_!K, L) \]

bifunctorially in $K$ and $L$.

**Proof.**
Choose a compactification $j : X \to \overline{X}$ over $Y$ and denote $\overline{f} : \overline{X} \to Y$ the structure morphism. Then we have

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ X(K, f^!L) & = \mathop{\mathrm{Hom}}\nolimits _ X(K, j^*\overline{f}{}^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_{\overline{X}})} (Rj_!K, \overline{f}{}^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_*Rj_!K, L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_!K, L) \end{align*}

The first equality follows immediately from the construction of $f^!$ in Section 48.16. By Lemma 48.17.6 we have $\overline{f}{}^!L$ in $D^+_{\textit{Coh}}(\mathcal{O}_{\overline{X}})$ hence the second equality follows from Lemma 48.30.2. Since $\overline{f}$ is proper the functor $\overline{f}{}^!$ is the right adjoint of pushforward by construction. This is why we have the third equality. The fourth equality holds because $Rf_! = Rf_* Rj_!$.
$\square$

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