Lemma 4.27.19. Let \mathcal{C} be a category and let S be a multiplicative system. The category of left fractions and the category of right fractions S^{-1}\mathcal{C} are canonically isomorphic.
Proof. Denote \mathcal{C}_{left}, \mathcal{C}_{right} the two categories of fractions. By the universal properties of Lemmas 4.27.8 and 4.27.16 we obtain functors \mathcal{C}_{left} \to \mathcal{C}_{right} and \mathcal{C}_{right} \to \mathcal{C}_{left}. By the uniqueness statement in the universal properties, these functors are each other's inverse. \square
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