I am quite confused of the wording on Hatcher's Algebraic Topology (p29-30) proof particularly this part:
What does he mean by "A unique lift $\tilde{F}: I \times \{ 0 \} \rightarrow \tilde{X}$ is obtained by an application of (a)"?
My understanding is $F: I \times \{0 \} \rightarrow X$ is a path, and by requiring $\tilde{F}(0 \times 0) = \tilde{x_0}$, where $\tilde{x_0} \in p^{-1}[F( 0 \times 0)]$ ($p$ is covering map), we obtain a unique path $\tilde{F}:I \times \{0\} \rightarrow \tilde{X}$. Is this right?