# Is toroidalization local?

Let $$f:X \to Y$$ be a surjective morphism of smooth projective varieties, $$D$$ be a simple normal crossings divisor on $$X$$ and $$U_Y \subset Y$$ be an open subset over which $$(X,D)$$ is log smooth (in the sense of the restriction of $$f$$ to any stratum is smooth over $$U$$). Then, I think by definition (if I was not mistaken about the definition), denoting $$U=f^{-1}U_Y$$, the morphism $$f|_U:(U,D|_U) \to U_Y$$ is toroidal.

Do we have modifications $$\pi: X' \to X,\phi: Y' \to Y$$ and an induced morphism $$f':(U' \subset X') \to (U_{Y'} \subset Y')$$

which satisfies:

1 $$f':(U' \subset X') \to (U_{Y'} \subset Y')$$ is toroidal.

2 $$\pi^{-1}D \subset X' \backslash U'$$.

3 $$\phi: U_{Y'} \to U_Y, \pi : f'^{-1} U_{Y'} \to U$$ are isomorphisms.

Roughly speaking, can we toroidalize varieties and morphisms only on $$$$non-toroidal" places?