 For a morphism $f: X \to Y$ in the category $\m V=\m V^{(qp)}_k$ or $\m V=\m V^{an}_c$, we consider a blow-up diagram $$\begin{CD} E @> i'>> Bl_{S}X' \\ @VV q' V  @VV q V \\ S @> i >> X' @> h >> X  @> f >> Y\:, \end{CD}$$ with $h$ proper and $i$ a closed embedding such that $f \circ h$ as well as $f \circ h \circ i$ are smooth. 