Let $A=l^\infty(\mathbb Z,\mathbb Z)$ be the abelian group of bounded sequences $\mathbb Z\to\mathbb Z$. Define a homomorphism $f\colon A\to A$ by $$f(a)(n)=a(2n)+a(2n+1),$$ for $a\in A$ and $n\in\mathbb Z$.
Consider the directed system $$A\xrightarrow{f}A\xrightarrow{f}A\xrightarrow{f}...$$
Question: What is the direct limit of this system?
Thoughts: The map $f$ is induced by "fusing together" two the $(2n)$-th copy of $\mathbb Z$ and the $(2n+1)$-th copy, i.e. $f$ is a direct sum of the homomorphisms $$A\supseteq\mathbb Z_{(2n)}\oplus\mathbb Z_{(2n+1)}\to\mathbb Z_{(n)}\subseteq A,$$ $$(x,y)\mapsto x+y,$$ where a subscript in parentheses denotes the index of a copy of $\mathbb Z$ in $A$. In particular, $\underbrace{f\circ f\circ\ldots\circ f}_{k\text{ times}}$ is a surjective map for any $k$. So I'm tempted to think that the direct limit is isomorphic to $A$, but I don't see a natural way to write down elements of the direct limit.