Example 4.22.6. Let $\mathcal{C}$ be a category. Let $(X_ n)$ and $(Y_ n)$ be inverse systems in $\mathcal{C}$ over $\mathbf{N}$ with the usual ordering. Picture:

$\ldots \to X_3 \to X_2 \to X_1 \quad \text{and}\quad \ldots \to Y_3 \to Y_2 \to Y_1$

Let $a : (X_ n) \to (Y_ n)$ be a morphism of pro-objects of $\mathcal{C}$. What does $a$ amount to? Well, for each $n \in \mathbf{N}$ there should exist an $m(n)$ and a morphism $a_ n : X_{m(n)} \to Y_ n$. These morphisms ought to agree in the following sense: for all $n' \geq n$ there exists an $m(n', n) \geq m(n'), m(n)$ such that the diagram

$\xymatrix{ X_{m(n, n')} \ar[rr] \ar[d] & & X_{m(n)} \ar[d]^{a_ n} \\ X_{m(n')} \ar[r]^{a_{n'}} & Y_{n'} \ar[r] & Y_ n }$

commutes. After replacing $m(n)$ by $\max _{k, l \leq n}\{ m(n, k), m(k, l)\}$ we see that we obtain $\ldots \geq m(3) \geq m(2) \geq m(1)$ and a commutative diagram

$\xymatrix{ \ldots \ar[r] & X_{m(3)} \ar[d]^{a_3} \ar[r] & X_{m(2)} \ar[d]^{a_2} \ar[r] & X_{m(1)} \ar[d]^{a_1} \\ \ldots \ar[r] & Y_3 \ar[r] & Y_2 \ar[r] & Y_1 }$

Given an increasing map $m' : \mathbf{N} \to \mathbf{N}$ with $m' \geq m$ and setting $a'_ i : X_{m'(i)} \to X_{m(i)} \to Y_ i$ the pair $(m', a')$ defines the same morphism of pro-systems. Conversely, given two pairs $(m_1, a_1)$ and $(m_1, a_2)$ as above then these define the same morphism of pro-objects if and only if we can find $m' \geq m_1, m_2$ such that $a'_1 = a'_2$.

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