Remark 4.22.5. Let $\mathcal{C}$ be a category. There exists a big category $\text{Pro-}\mathcal{C}$ of pro-objects of $\mathcal{C}$. Namely, if $F : \mathcal{I} \to \mathcal{C}$ and $G : \mathcal{J} \to \mathcal{C}$ are cofiltered diagrams in $\mathcal{C}$, then we can define

$\mathop{Mor}\nolimits _{\text{Pro-}\mathcal{C}}(F, G) = \mathop{\mathrm{lim}}\nolimits _ j \mathop{\mathrm{colim}}\nolimits _ i \mathop{Mor}\nolimits _\mathcal {C}(F(i), G(j)).$

There is a canonical functor $\mathcal{C} \to \text{Pro-}\mathcal{C}$ which maps $X$ to the constant system on $X$. This is a fully faithful embedding. In this language one sees that a diagram $F$ is essentially constant if and only if $F$ is isomorphic to a constant system. If we ever need this material, then we will formulate this into a lemma and prove it here.

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