Let $S_{1},S_{2}$ be quasi-projective varieties over $\mathbb{C}$ of dimension $2$ with only isolated, quotient singularities. Suppose $\phi : S_{1} \rightarrow S_{2}$ is a morphism that is generically 1-1.
Let $\tilde{S_{i}}$ denote minimal (i.e. exceptional divisors contain no $-1$ curves) resolutions of the $S_{i}$, can we lift $\phi$ uniquely to a morphism $\tilde{\phi} : \tilde{S_{1}} \rightarrow \tilde{S_{2}}$?