Lemma 4.26.11. Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system.

1. The relation on pairs defined above is an equivalence relation.

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

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

Proof. This lemma is dual to Lemma 4.26.2. It follows formally from that lemma by replacing $\mathcal{C}$ by its opposite category in which $S$ is a left multiplicative system. $\square$

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