Lemma 15.92.1. Let $A$ be a ring and let $I \subset A$ be an ideal. The category $\mathcal{C}$ of derived complete modules is abelian, has arbitrary limits, and the inclusion functor $F : \mathcal{C} \to \text{Mod}_ A$ is exact and commutes with limits. If $I$ is finitely generated, then $\mathcal{C}$ has arbitrary colimits and $F$ has a left adjoint

**Proof.**
This summarizes the discussion above.
$\square$

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