This is exercise 1.3.8 in Hatcher:
Let $\tilde{X}$ and $\tilde{Y}$ be simply-connected covering spaces of path connected, locally path-connected spaces $X$ and $Y$. Show that if $X\simeq Y$ then $\tilde{X}\simeq \tilde{Y}$.
I tried applying the lifting criterion, but I seem to be hitting a dead end. Any help would be much appreciated.