Lemma 4.22.3 (0G2V)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 4: Categories
Section 4.22: Essentially constant systems
Lemma 4.22.3 (cite)

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Lemma 4.22.3. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. If $\mathcal{I}$ is filtered and $M$ is essentially constant as an ind-object, then $X = \mathop{\mathrm{colim}}\nolimits M_ i$ exists and $M$ is essentially constant with value $X$ . If $\mathcal{I}$ is cofiltered and $M$ is essentially constant as a pro-object, then $X = \mathop{\mathrm{lim}}\nolimits M_ i$ exists and $M$ is essentially constant with value $X$ .

Proof. Omitted. This is a good exercise in the definitions. $\square$

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