The Stacks project

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 excercise in the definitions. $\square$


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