Understanding Direct Limits: $\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})$

[cohomology]

I'm trying to understand direct limits so consider the direct limit

$\lim_\rightarrow (\mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \mathbb{Z} \rightarrow \cdots)$
where each map is multiplication by 2.
I concluded that the solution is $\mathbb{Z}\sqcup\coprod(\mathbb{Z}-2\mathbb{Z})$. Is this correct?

 

[yyat]

OK, note first that the direct limit of groups should again be a group.
In this particular example it is isomorphic to $$\mathbb{Z}[1/2]$$, i.e. the multiplicative group of rational numbers of the form $$a/2^b$$, where a,b are integers, $$b\geq0$$.