Proposition 52.6.12. Let (\mathcal{C}, \mathcal{O}) be a ringed site. Let \mathcal{I} \subset \mathcal{O} be a finite type sheaf of ideals. There exists a left adjoint to the inclusion functor D_{comp}(\mathcal{O}) \to D(\mathcal{O}).
Proof. Let K \to \mathcal{O} in D(\mathcal{O}) be as constructed in Lemma 52.6.11. Let E \in D(\mathcal{O}). Then E^\wedge = R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (K, E) together with the map E \to E^\wedge will do the job. Namely, locally on the site \mathcal{C} we recover the adjoint of Lemma 52.6.8. This shows that E^\wedge is always derived complete and that E \to E^\wedge is an isomorphism if E is derived complete. \square
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