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using universal formula for free abelian group and set theory of a group
I’m trying to understand (or learn from) the proof of Theorem 2 here:
It says that if we have a relation of the form $a=c\cdot(x-y)$ with $x,y\in\mathbb{R}$, then for any subgroup $H$ of the additive group of $\mathbb{R}$, the set $H-\{0\}$ has a unique (up to additive inverse) $c$-expansion if and only if $\{x,y\}$ is not a subset of $H$. I understand this to mean that any element of $H-\{0\}$ can be written uniquely as $u_1\cdot x+u_2\cdot y$ for $u_1,u_2\in\mathbb{Z}$.
I also understand that if we use the universal property for the free abelian group on $2$ elements, and the map $f:a\mapsto(a,0)$, then for any subgroup $H$ of $\mathbb{Z}^2$, the set $H-\{0\}$ has a unique (up to additive inverse) $c$-expansion if and only if $\{x,y\}$ is not a subset of $H$. However, I don’t understand the specific properties of this universal map, nor the universal property. In particular, how do we make sure $f$ satisfies the universal property for the free abelian group?
A:
If $A= \langle a \rangle \subseteq \mathbb{Z}^2$ is a subgroup then $\bar{A}= \langle a,0 \rangle$ is a subgroup of $\mathbb{Z}^
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