All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). Then locally on $X$ and $Y$ we have factorizations
$X \overset{f}{\rightarrow} S \times \mathbb{A}^n \rightarrow S$
$Y \overset{g}{\rightarrow} S \times \mathbb{A}^m \rightarrow S$
where $f$ and $g$ are etale. My question is whether it is always possible to get a local factorization of $\iota $ as
$\require{AMScd}$ \begin{CD} X @>{\iota}>> Y\\ @VVV @VVV \\ S \times \mathbb{A}^n @>{\iota'}>> S \times \mathbb{A}^m \end{CD}
where $\iota'$ is defined by $ x_{n+1}= \dots x_{m} = 0$? Moreover, if this is possible can we set things up so that the diagram is cartesian?
