The Stacks project

Proof. By uniqueness of left duals (Categories, Remark 4.43.7) we get the final statement provided we show that $\mathcal{F}^\bullet$ is as stated. Let $\mathcal{G}^\bullet , \eta , \epsilon$ be a left dual. Write $\eta = \sum \eta _ n$ and $\epsilon = \sum \epsilon _ n$ where $\eta _ n : \mathcal{O}_ X \to \mathcal{F}^ n \otimes _{\mathcal{O}_ X} \mathcal{G}^{-n}$ and $\epsilon _ n : \mathcal{G}^{-n} \otimes _{\mathcal{O}_ X} \mathcal{F}^ n \to \mathcal{O}_ X$. Since $(1 \otimes \epsilon ) \circ (\eta \otimes 1) = \text{id}_{\mathcal{F}^\bullet }$ and $(\epsilon \otimes 1) \circ (1 \otimes \eta ) = \text{id}_{\mathcal{G}^\bullet }$ by Categories, Definition 4.43.5 we see immediately that we have $(1 \otimes \epsilon _ n) \circ (\eta _ n \otimes 1) = \text{id}_{\mathcal{F}^ n}$ and $(\epsilon _ n \otimes 1) \circ (1 \otimes \eta _ n) = \text{id}_{\mathcal{G}^{-n}}$. Hence we see that $\mathcal{F}^ n$ is locally a direct summand of a finite free $\mathcal{O}_ X$-module by Modules, Lemma 17.18.2. Since the sum $\eta = \sum \eta _ n$ is locally finite, we conclude that $\mathcal{F}^\bullet$ is locally bounded. $\square$