Lemma 56.7.2. Let $X$ and $Y$ be Noetherian schemes. Let $F : \textit{Coh}(\mathcal{O}_ X) \to \textit{Coh}(\mathcal{O}_ Y)$ be an equivalence of categories. Then there is an isomorphism $f : Y \to X$ and an invertible $\mathcal{O}_ Y$-module $\mathcal{L}$ such that $F(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}$.

**Proof.**
By Lemma 56.7.1 we obtain a unique functor $F' : \mathit{QCoh}(\mathcal{O}_ X) \to \mathit{QCoh}(\mathcal{O}_ Y)$ extending $F$. The same is true for the quasi-inverse of $F$ and by the uniqueness we conclude that $F'$ is an equivalence. By Proposition 56.6.6 we find an isomorphism $f : Y \to X$ and an invertible $\mathcal{O}_ Y$-module $\mathcal{L}$ such that $F'(\mathcal{F}) = f^*\mathcal{F} \otimes \mathcal{L}$. Then $f$ and $\mathcal{L}$ work for $F$ as well.
$\square$

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