Definition 4.27.12. Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\mathcal{C}$. Given any morphism $f : X' \to Y$ in $\mathcal{C}$ and any morphism $s : X' \to X$ in $S$, we denote by $f s^{-1}$ the equivalence class of the pair $(f : X' \to Y, s : X' \to X)$. This is a morphism from $X$ to $Y$ in $S^{-1} \mathcal{C}$.

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