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The Stacks project

Lemma 21.19.1. Let f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') be a morphism of ringed topoi. The functor Rf_* defined above and the functor Lf^* defined in Lemma 21.18.2 are adjoint:

\mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O})}(Lf^*\mathcal{G}^\bullet , \mathcal{F}^\bullet ) = \mathop{\mathrm{Hom}}\nolimits _{D(\mathcal{O}')}(\mathcal{G}^\bullet , Rf_*\mathcal{F}^\bullet )

bifunctorially in \mathcal{F}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O})) and \mathcal{G}^\bullet \in \mathop{\mathrm{Ob}}\nolimits (D(\mathcal{O}')).

Proof. This follows formally from the fact that Rf_* and Lf^* exist, see Derived Categories, Lemma 13.30.3. \square


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