In Hatcher's book,the lifting criterion is stated as following:
Suppose given a covering space $p: (\tilde{X},\tilde{x}_0) \rightarrow (X,x_0)$ and a map $f: (Y,y_0) \rightarrow (X,x_0)$ with $Y$ path-connected and locally path-connected. Then a lift $\tilde{f}: (Y,y_0) \rightarrow (\tilde{X},\tilde{x}_0)$ of $f$ exists iff $f_*(\pi_1 (Y,y_0)) \subset p_*(\pi_1(\tilde{X},\tilde{x}_0))$.
The proof of "only if" part is obvious.
I met with some problems when reading the proof the "if" part.
For any $y\in Y$,let $\gamma$ be a path in $Y$ from $y_0$ to $y$, then we can define $\tilde{f}(y)=\widetilde{f\gamma}(1)$.We need to show that the definition is well defined.Let $\gamma^{'}$ be another path in $Y$ from $y_0$ to $y$.Then $(f\gamma')\cdot(\overline{f\gamma})$ is a loop $h_0$ at $x_0$ with $[h_0]\in f_*(\pi_1 (Y,y_0)) \subset p_*(\pi_1(\tilde{X},\tilde{x_0}))$.*This means that there is a homotopy $h_t$ of $h_0$ to a loop $h_1$ that lifts to a loop $\tilde{h}_1$ in $\tilde{X}$ based at $x_0$.*How to deduce the above conclusion?
Applying the covering homotopy property to $h_t$ to get a lifting $\tilde{h}_t$.Since $\tilde{h}_1$ is a loop at $\tilde{x}_0$,so is $\tilde{h}_0$ .(Why?)
Best Answer
Both of your questions can be answered by one simple fact that the elements of $p_*(\pi_1(\widetilde X, \widetilde x_0))$ precisely the classes of those loops in $(X, x_0)$ which lift to a loop in $(\widetilde X, \widetilde x_0)$. In fact this result is presented in Hatcher (the relevant part is in bold):
The proof is given in Hatcher, but I present the proof of the part in bold for completeness.
Suppose $[\gamma] \in p_*(\pi_1(\widetilde X, \widetilde x_0))$, then $[\gamma] = p_*[\widetilde \gamma_1]$, for some loop $\widetilde \gamma_1$ in $(\widetilde X, \widetilde x_0)$. So, we have $\gamma \simeq p \circ \widetilde \gamma_1 = \gamma_1$ (say). $\gamma$ and $\gamma_1$ are homotopic as paths, so their liftings will also be homotopic as paths (by homotopy lifting property), that is, $\widetilde \gamma_1 \simeq \widetilde \gamma$. Now, since $\widetilde \gamma_1$ is a loop, so, is $\widetilde \gamma$.
Conversely, suppose $\gamma$ is a loop in $(X, x_0)$ that lifts to a loop $\widetilde \gamma$ in $(\widetilde X, \widetilde x_0)$, this means that $\gamma = p \circ \widetilde \gamma$ or $[\gamma] = p_*[\widetilde\gamma]$. So, $[\gamma] \in p_*(\pi_1(\widetilde X, \widetilde x_0))$.
Now, coming back to your questions, note that since $[h_0] \in p_*(\pi_1(\widetilde X, \widetilde x_0))$, $h_0$ lifts to loop.