Different coordinates of Witt vectors

In Kedlayas lecture notes (https://kskedlaya.org/prismatic/sec_overview.html) about prismatic cohomology he introduces the ring of Witt vectors using $\delta$-rings via the fact that the Witt vector functor $W$ is the right adjoint to the forgetful functor $\mathbf{Ring}_{\delta}\to \mathbf{Ring}$. Using the adjunction properties he shows in section 3.1 that $W(A)$ looks like the infinite product ring $A^{\mathbb{N}}$ on the level of sets. From this he deduces the existence of a unique expansion of each element of $W(A)$ as $y=(y_0,y_1,\dots)$ which he calls the $y$-coordinates. In Lemma 3.1.3 he proves the existence of another set of generators, called the $x$-coordinates of the free $\delta$-ring $\mathbb{Z}\{y\}$.
My questions are: 1. Why does it suffice to prove the existence of these coordinates in $\mathbb{Z}\{y\}$ (how do we transfer this to arbitrary rings $W(A)$?)
2. What would be an explicit example of the y-coordinates, the x-coordinates and the ghost coordinates in comparison?

I would be grateful for any suggestions.