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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