Krdytkyu

In studying to write an expository paper in representation theory,
I am reading Abelian Categories with Applications to Rings and Modules
by Popescu and I have not been able to figure out something which
appears to go without saying in the book.

In Section 5.1, Theorem 1.3 (Azumaya) says that, in an AB5 category, decomposition
of an object into indecomposable objects with local endomorphism rings
is unique in the Krull-Remak-Schmidt-Azumaya sense (i.e. up to reordering
and isomorphism).

The following characterization of the condition AB5 is used in the
proof (at the moment, I do not have the original book, but only the
notes I have made while reading, so I might be using a different notation).
Let $mathscr{A}$ be an abelian category which has arbitrary coproducts.
Let $left{ A_{i}right} _{iin I}$ be a set of objects of $mathscr{A}$
and let
$$
iota_{i}^{I}:A_{i}tocoprod_{iin I}A_{i}
$$
denote the coprojection for $iin I$. For
any finite subset $F$ of the set $I$, let $iota_{f}^{F}$ denote
the coprojection of $A_{f}$ into $coprod_{fin F}A_{f}$, and let
$A_{F}$ denote the source of the image (that is the image itself
if an image is thought of as an object and not an arrow) of the canonical
arrow
$$
u_{F}:coprod_{fin F}A_{f}tocoprod_{iin I}A_{i}
$$
defined such that
$$
u_{F}iota_{f}^{F}=iota_{f}^{I}
$$
for all $fin F$. If for every subobject $A$ of the object $coprod_{iin I}A_{i}$
holds the equality
$$
A=sum_{Fin T}left(Acap A_{F}right),
$$
with $T$ denoting the set of all finite subsets of $I$, it is said
that $mathscr{A}$ verifies the condition AB5 or that it is an AB5
category.

In the same section, Theorem 1.4 says that, in an Abelian category
in which every double chain is stationary, every object has a unique
(in the Krull-Remak-Schmidt-Azumaya sense) decomposition into finitely many
indecomposable objects with local endomorphism rings.

By double chain, one means
$$
left(A_{n}{rightleftarrows}A_{n+1}right)_{ninmathbb{N}},
$$
with $A_{n}$ being objects, $i_{n}:A_{n+1}to A_{n}$ being monomophisms and $p_{n}:A_{n}to A_{n+1}$
being epimorphisms. The chain is stationary if there is $n_{0}inmathbb{N}$
such that $i_{n}$ and $p_{n}$ are isomorphisms for all $ngeq n_{0}$.

For the proof of the uniqueness part of Theorem 1.4, it is said in
the book that Theorem 1.3 is used. No proof is given, though.

My question: How does one prove that the condition for using Theorem
1.3 is satisfied in Theorem 1.4, namely that a category described
in Theorem 1.4 indeed verifies the aforementioned condition AB5 needed
for Theorem 1.3?

Thank you for your time and attention!