Lemma 56.2.1. Let $A$ be a ring. Let $\mathcal{B}$ be a category having filtered colimits. Let $F : \text{Mod}^{fp}_ A \to \mathcal{B}$ be a functor. Then $F$ extends uniquely to a functor $F' : \text{Mod}_ A \to \mathcal{B}$ which commutes with filtered colimits.

Proof. This follows from Categories, Lemma 4.26.2. To see that the lemma applies observe that finitely presented $A$-modules are categorically compact objects of $\text{Mod}_ A$ by Algebra, Lemma 10.11.4. Also, every $A$-module is a filtered colimit of finitely presented $A$-modules by Algebra, Lemma 10.11.3. $\square$

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