We follow the argument in [NXY19, Proposition 5.4].
Since the source and the target have same dimension and are integral, it is enough to check that $f$ is a closed immersion.
If $\mathscr{J}$ is the largest ideal of definition of $\widehat{\mathscr{N}_{/Z}}$, i.e. the defining ideal of $Z\subset\mathscr{N}$, then $f^{*}\mathscr{J}=\mathscr{I}_{Z}$ is the largest ideal of definition of $\widehat{\mathscr{X}_{/Z}}$. Indeed, since $Z$ is cut out inside $\mathscr{X}$ by the $D_{j}$ for $j\in J$, the ideal $\mathscr{I}_{Z}$ is locally generated by the $s_{j}$ for $j\in J$; the same reasoning shows that $\mathscr{J}$ is locally generated by the $\chi^{\varepsilon_{j}}$. The equality $f^{*}\mathscr{J}=\mathscr{I}_{Z}$ now follows directly from the local definition of $f$.
We use [Gro61, 4.8.10] and the fact that $f$ induces an isomorphism on the reductions to infer that $f$ is a closed immersion, and thus an isomorphism by equality of dimensions. ∎