Suppose we have a morphism $\phi : S_{1} \rightarrow S_{2}$, between quasi-projective varieties of dimension $2$ over $\mathbb{C}$ with at worst quotient singularities. Suppose furthermore that $\phi$ is an isomorphism on a Zariski open subset.

Let $\tilde{S_{i}}$ denote minimal resolutions of the $S_{i}$ (exceptional divisors contain no $-1$ curves). Is it possible to lift $\phi$ uniquely to a morphism $\tilde{\phi}: \tilde{S_{1}} \rightarrow \tilde{S_{2}}$?

I asked this question already on math.stackexchange (https://math.stackexchange.com/questions/2735590/lifting-a-generically-1-1-map-to-resolutions)