I'm struggling understanding a small sentece from Hatcher's Algebraic topology book (available online for free).
In page 70 Hatcher wants to reconstruct the covering $p:\tilde X\to X$ from the associated action $\pi_1(X,x_0)\to F=p^{-1}(x_0)$ assuming that $X$ is path-connected, locally path-connected, and semilocally path connected. So he takes the universal covering space $\tilde X_0$ constructed some pages before that consists of the homotopy classes of paths starting at $x_0$. He then defines a map $h:\tilde X_0\times F\to \tilde X$ as $h([\gamma],\tilde x_0)=\tilde \gamma_{\tilde x_0}(1)$ where $\tilde \gamma_{\tilde x_0}$ is the unique lift of $\gamma$ starting at $\tilde x_0$. Then he proves that $h$ is continuous, and in fact, a local homeomorphism. Details about the covering space $\tilde X_0$ are given in page 64.
After proving this he says: It is obvious that $h$ is surjective since $X$ is path-connected. This is the sentence that I don't understand. So for $\tilde a\in \tilde X$ one has to prove that there is $\gamma$ path in $X$ starting at $x_0$ and $\tilde x_0\in F$ such that $h([\gamma],\tilde x_0)=\tilde \gamma_{\tilde x_0}(1)=\tilde a$, this is the same as saying that given $\tilde a$ there is $\tilde x_0\in F$ and a path $\tilde \gamma$ from $\tilde x_0$ to $\tilde a$ because then we can simply take $\gamma=p\tilde \gamma$. I don't get how $X$ being path-connected comes into play.