Lemma 4.27.11. Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system.

The relation on pairs defined above is an equivalence relation.

The composition rule given above is well defined on equivalence classes.

Composition is associative (and the identity morphisms satisfy the identity axioms), and hence $S^{-1}\mathcal{C}$ is a category.

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