I want to use the following resolution of singularity statment as found in Soule et al, Lectures on Arakelov Geometry, p. 40:
$Y$ is a separated algebraic variety of finite type over $\mathbb{C}$, $Z$ a Zariski closed subset containing the singular locus of $Y$. Then there exists a proper map $ \pi: \tilde{Y} \longrightarrow Y$ such that
(1) $\tilde{Y}$ is smooth
(2) $E:= \pi^{-1}(Z)$ is a DNC
(3) $\pi: \tilde{Y} \setminus E \longrightarrow Y \setminus Z$ is an Iso.
$\textbf{Additionally}$ I would like to have a functoriality property like
(4) (Functoriality) Given a smooth morphism $f: X \longrightarrow Y$ then there is a resolution of singularities of $X$ along $f^{-1}(Z)$ as above such that
$\require{AMScd}$ \begin{CD} \tilde{X} @>\tilde{f}>> \tilde{Y}\\ @V V V @VV V\\ X @>>f> Y \end{CD}
commutes and $\tilde{f}$ smooth.
No reference I went through gave this statement explicitely. They always proof a Principalization statment and deduce the case $Z= Y_{sing}$ as a Corollary.
I would like to avoid going to deep into the theory and use this as a black box. Do you know any reference? Or any way this follows direct from a Principalization Theorem as the one in Kollar's Lecture notes?
Thank you very much!
