What is the induced map on group completions defined by coherence sequences? This is primarily motivated by Atiyah and MacDonald's discussion of completions on page 102-103 of Introduction to Commutative Algebra.
An abelian group $G$ can be topologized by taking a sequence of subgroups
$$
G=G_0\supset G_1\supset G_2\supset\cdots
$$
and defining this to a be fundamental system of neighborhoods of $0$. Then the completion $\hat{G}$ may be defined to be set of coherent sequences $(\xi_i)$ with pointwise addition, where $\xi_i\in G/G_i$, and a sequence is said to be coherent if $\theta_{i+1}\xi_{i+1}=\xi_i$, where $\theta_{i+1}\colon G/G_{i+1}\to G/G_i$ is simply the map $g+G_{i+1}\mapsto g+G_i$.
My question is, if $f\colon G\to H$ is a continuous map of abelian groups topologized in the above way, what is the induced map $\hat{f}\colon\hat{G}\to\hat{H}$? In the alternative definition of a completion using (equivalence classes of) Cauchy sequences instead of coherent sequences, you can just apply the map pointwise to a Cauchy sequence, but to do so here seems like you would need $f$ to induce morphisms
$$
G/G_i\to H/H_i : g+G_i\mapsto f(g)+H_i
$$
for all $i$, but this seems like you would need $f(G_i)\subseteq H_i$ for all $i$, which seems like an extra condition.
 A: You shouldn't think of coherent sequences as sequences that you want to apply maps to pointwise.  Instead, you should think of a sequence $(\xi_i)$ as representing an element $\xi\in\hat{G}$ by telling you what $\xi$ is mod $G_i$ for each $i$.  An element of $\hat{G}$ is like an element of $G$, but instead of having an actual element of $G$, we only know its values mod $G_i$ for each $i$ (and these values are compatible with each other).
So, if we want to define $\hat{f}(\xi)$, what do we do?  Well, we need to know what $\hat{f}(\xi)$ is mod $H_i$ for each $i$, since that's what it takes to give an element of $\hat{H}$.  Now by continuity of $f$, there is some $j$ such that $f(G_j)\subseteq H_i$.  This means that if we know the value of $\xi$ mod $G_j$, that should be enough to know "$f(\xi)$" mod $H_i$, since our error in the input is in $G_j$ so the error in the output will be in $H_i$.  So to define the value of $\hat{f}(\xi)$ mod $H_i$, we should just apply $f$ to the value of $\xi$ mod $G_j$.
That is, the $i$th term of the sequence $\hat{f}(\xi)$ is defined as follows.  Let $j$ be such that $f(G_j)\subseteq H_i$, and let the $j$th term of $\xi$ be $g_j+G_j$.  Then the $i$th term of $\hat{f}(\xi)$ is $f(g_j)+H_i$.  (This does not depend on the choice of representative $g_j$ since $f(G_j)\subseteq H_i$, and does not depend on the choice of $j$ with this property since $\xi$ is coherent.)  I will leave it to you to check that this definition actually defines a coherent sequence $\hat{f}(\xi)$ and makes $\hat{f}$ a continuous homomorphism.
