Remark 4.27.15. Let $\mathcal{C}$ be a category. Let $S$ be a right multiplicative system. Given an object $X$ of $\mathcal{C}$ we denote $S/X$ the category whose objects are $s : X' \to X$ with $s \in S$ and whose morphisms are commutative diagrams

where $a : X' \to X''$ is arbitrary. The category $S/X$ is cofiltered (see Definition 4.20.1). (This is dual to the corresponding statement in Remark 4.27.7.) Now the combined results of Lemmas 4.27.13 and 4.27.14 tell us that

This formula expressing morphisms in $S^{-1}\mathcal{C}$ as a filtered colimit of morphisms in $\mathcal{C}$ is occasionally useful.

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