Remark 48.32.4. Let $\mathcal{C}$ be a category. Suppose given an inverse system

of inverse systems in the category of pro-objects of $\mathcal{C}$. In other words, the arrows $\alpha _ i$ are morphisms of pro-objects. By Categories, Example 4.22.6 we can represent each $\alpha _ i$ by a pair $(m_ i, a_ i)$ where $m_ i : \mathbf{N} \to \mathbf{N}$ is an increasing function and $a_{i, n} : M_{i, m_ i(n)} \to M_{i - 1, n}$ is a morphism of $\mathcal{C}$ making the diagrams

commute. By replacing $m_ i(n)$ by $\max (n, m_ i(n))$ and adjusting the morphisms $a_ i(n)$ accordingly (as in the example referenced) we may assume that $m_ i(n) \geq n$. In this situation consider the inverse system

with general term

For any object $N$ of $\mathcal{C}$ we have

We omit the details. In other words, we see that the inverse system $(M_ k)$ has the property

This property determines the inverse system $(M_ k)$ up to pro-isomorphism by the discussion in Categories, Remark 4.22.7. In this way we can turn certain inverse systems in $\text{Pro-}\mathcal{C}$ into pro-objects with countable index categories.

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