The Stacks project

Lemma 4.22.10. Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with cofiltered index category $\mathcal{I}$. The following are equivalent

1. $M$ is an essentially constant pro-object,

2. there exists a cone $(X, \{ X \to M_ i\} )$ such that for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W)$ is bijective,

3. $X = \mathop{\mathrm{lim}}\nolimits _ i M_ i$ exists and for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W) \to \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W)$ is bijective, and

4. there exists an $i$ in $\mathcal{I}$ and a morphism $M_ i \to X$ such that for any $W$ in $\mathcal{C}$ the map $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(X, W) \to \mathop{\mathrm{colim}}\nolimits _{i \in \mathcal{I}^{opp}} \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ i, W)$ is bijective.

In cases (2), (3), and (4) the value of the essentially constant system is $X$.