The set $5^{-\infty}\mathbb{Z}$ is a colimit

Hint: $5^{-n}\Bbb{Z}$ here means $\{5^{-n} x \mid x \in \Bbb{Z}\} \subseteq \Bbb{Q}$. The arrow $5^{-n}\Bbb{Z} \to 5^{-n-1}\Bbb{Z}$ is the inclusion function $x \mapsto x$ (a multiple of $1/5$ is also a multiple of $1/25$ etc.). The colimit construction gives that the elements of the colimit are represented by finite sequences $(x_1, \ldots x_n)$ of integers with $x_n \neq 0$ (or $n = 1$ and $x_n = 0$) under an equivalence relations that means each such sequence can be identified with $x_n/5^n$.