Let $X$ and $Y$ be topological spaces.

(i) Show that a map $f: X \rightarrow Y$ is a homotopy equivalence if there exist maps $g, h: Y \rightarrow X$ such that $f g \simeq 1_{Y}$ and $h f \simeq 1_{X}$. More generally, show that a map $f: X \rightarrow Y$ is a homotopy equivalence if there exist maps $g, h: Y \rightarrow X$ such that $f g$ and $h f$ are homotopy equivalences.

(ii) Suppose that $\tilde{X}$ and $\tilde{Y}$ are universal covering spaces of the path-connected, locally path-connected spaces $X$ and $Y$. Using path-lifting properties, show that if $X \simeq Y$ then $\tilde{X} \simeq \tilde{Y}$.