the **universal property** of a good uniformizer, which says that given a variation $t\mapsto\beta_t$ and a good uniformizer $\tau\mapsto \alpha_\tau$ such that $\beta_{t_0}$ is isomorphic to $\alpha_{\tau_0}$ via $\phi_0$, then there exist unique $t\mapsto \tau(t)$ with $\tau(t_0)=\tau_0$ and a unique family of isomorphisms $\phi_t:\beta_t\mapsto \alpha_{\tau(t)}$ extending $\phi_{t_0}=\phi_0$ (see [@PolyfoldGW] 3.13 and for the moment we postpone the sc-smoothness content of this important theorem and revisit it shortly).

To have perturbation theory (which boils down to the contraction mapping theorem on a Banach space somewhere), we need to consider the stable map space locally as a Banach space, hence being of finite regularity (for example, $H^{3,\delta}$ with $\delta>0$ in a holomorphic polar coordinate around nodes and $H^3_{\text{loc}}$ around points elsewhere). The aforementioned $\tau(t)$ in the universal property is a composition of maps, one of which is a domain reparametrization and nowhere $C^1$. This together with the forms of other maps appearing in $\tau(t)$ means that $\tau(t)$ is nowhere $C^1$, hence in particular we only have continuous coordinate change on $B(Q,\omega)$ for this choice of smooth structure $H^{3,\delta}$, or any other classical choice.

For clarity, we abstractly write $f:E_0\to F_0$ for a localized domain reparametrization $(u,a)\mapsto u\circ\phi_a$ where $a\mapsto \phi_a$ is a 