[@Polyfoldanalysis] 2.1 was then obtained when they tried to relate it to the classical smoothness. This gives a glimpse of the analytic prowess of HWZ. $sc^k$ can be defined inductively via $T^{k-1}f$. Being **sc-smooth** just means being $sc^k$ for all $k$. We can regard sc-smoothness for a map $f$ between two sc-Banach spaces as a property for the map $f$ between 0-levels and its actual sc-smoothness is then specified by the information from the sc-structures of domain and target sc-Banach spaces.

An sc-open set $O$ in an sc-Banach space $E$ means $(O_m)_{m\in\mathbb{N}_0}$ such that $O_0$ is an open set in $E_0$ and $O_m=O_0\cap E_m$. Following HWZ, we often use $E$ to mean both level-0 Banach space $E_0$ as well as $(E_m)_m$. It is often clear in each context which meaning is being used; and when there are two possible meanings, the contents are the same. For example, $O\subset E$ can mean both the inclusion between the 0-th levels and the levelwise inclusion, and both contents are equivalent by definition. This new sc-smooth notion gives rise to new a local model $O:=r(U)$, where $r:U\to U$ is an sc-smooth map between the same sc-open set $U$ in sc-Banach $E$ and idempotent ($r\circ 