How do you turn an abstract set constraint into equality constraints?

I am reading from Nonlinear Programming by Bertsekas, and in the section on ADMM, he says:

Consider the problem $$\text{min} \sum _{i=1} ^ m f_i(x)$$ $$\text{s.t. }x \in \cap _{i = 1}^m X_i,$$ where $$f_i : \mathbb{R}^n \to \mathbb{R}$$ are convex functions and $$X_i$$ are closed, convex sets with nonempty intersection. We can reformulate this as an equality constrained problem, by introducing additional artificial variables $$z_i, i = 1,..., m$$ and the equality constraints $$x = z_i$$: $$\text{min} \sum _{i=1} ^ m f_i(z_i)$$ $$\text{s.t.} \hspace{0.3 cm} x = z_i, \hspace{0.6cm} z_i \in X_i,\hspace{0.6cm} i = 1, ..., m$$

I don't quite understand this transformation. I see that we have $$m$$ different functions, and for each function $$f_i$$ its input must come from a (possibly) different set $$X_i$$. That makes sense. But what does $$x$$ mean in this second problem? My only idea is that $$x$$ doesn't really stand for anything; it is just there as a short way of saying $$z_1 = z_2 = ... = z_m$$. Is this correct?

Thank you very much!

Yes, $$x$$ is the common value of the $$z_i$$s. This idea of introducing multiple copies of a variable is known as Lagrangian Decomposition. The $$x=z_i$$ equalities are linking constraints for what otherwise would decompose into $$m$$ disjoint subproblems.