Suppose $(X,d)$ is a separable metric space with $d\leq 1$, and let $(x_n)$ and $(x'_n)$ be two dense sequences. Define embeddings $$\varphi,\varphi':X\to[0,1]^\mathbb{N}$$ by putting $\varphi(x)=(d(x,x_n))_{n}$ and $\varphi'(x)=(d(x,x'_n))_{n}$ respectively. Then $\mathcal{X}=\overline{\varphi(X)}$ and $\mathcal{X}'=\overline{\varphi'(X)}$ are two metrizable compactifications of $X$.
It seems to me that $\mathcal{X}$ and $\mathcal{X}'$ always homeomorphic. In order to define a maps $\mathcal{X}\xrightarrow\theta\mathcal{X}'$ and $\mathcal{X}'\xrightarrow{\theta'}\mathcal{X}$ we define $\theta$ on $\varphi(X)$ as $\varphi'\circ\varphi^{-1}$ (I'm cutting corners here), and extend it by uniform continuity (and similarly for $\theta'$). I haven't checked the details, but this will produce (uniformly) continuous maps $\mathcal{X}\xrightarrow\theta\mathcal{X}'$ and $\mathcal{X}'\xrightarrow{\theta'}\mathcal{X}$ satisfying $\theta\circ\varphi=\varphi'$ and $\theta'\circ\varphi'=\varphi$ so that $\theta\circ\theta'$ coïncides with the identity function on the dense subset $\varphi(X)\subset\mathcal{X}$, and $\theta'\circ\theta$ coïncides with the identity function on the dense subset $\varphi'(X)\subset\mathcal{X}'$. And thus we should get that $\theta$ and $\theta'$ are inverse homeomorphisms of one another.
Question : Is the above reasoning sound, and if so, what is the name of this compactification ? Does it have a universal property ? Maybe something similar to the Stone-Cech compactification but in the category of compact metric spaces ?