Let $(X,d)$ be a metric space. Let $({X_1}^*, {d_1}^*)$ and $({X_2}^*, {d_2}^*)$ be completions of $(X,d)$ such that $\phi_1:X\rightarrow {X_1}^*$ and $\phi_2:X\rightarrow {X_2}^*$ are isometries. ($\phi_1[X]$ and $\phi_2[X]$ are dense in ${X_1}^*$ and ${X_2}^*$ respectively)
Then, there exists a unique bijective isometry $f:{X_1}^* \rightarrow {X_2}^*$ such that $f\circ \phi_1 = \phi_2$.
Here, let $\phi_1=\phi_2$. It doesn't seem to me that ${X_1}^* = {X_2}^*$.
What is 'unique' this theorem referring to?