The Stacks project

Lemma 4.27.13. Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\mathcal{C}$. Given any finite collection $g_ i : X \to Y_ i$ of morphisms of $S^{-1}\mathcal{C}$ (indexed by $i$), we can find an element $s : X' \to X$ of $S$ and a family of morphisms $f_ i : X' \to Y_ i$ of $\mathcal{C}$ such that $g_ i$ is the equivalence class of the pair $(f_ i : X' \to Y_ i, s : X' \to X)$.

Proof. This lemma is the dual of Lemma 4.27.5 and follows formally from that lemma by replacing all categories in sight by their opposites. $\square$

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