The homotopy lifting property says:
Suppose $p: \tilde X \to X$ is a covering map. Suppose $F: Y \times I \to X$ is a homotopy between $f_0$ and $f_1$, and $\tilde f_0$ is a lift of $f_0$. Then there exists a unique homotopy $\tilde F: Y \times I \to \tilde X$ such that $\tilde F(s,0)=\tilde f_0(s)$ and $p \circ \tilde F = F$.
Does this mean that the homotopy $\tilde F$ is a homotopy between $\tilde f_0$ and $\tilde f_1$, where $\tilde f_0$ is a lift of $f_0$ and $\tilde f_1$ is a lift of $f_1$?
Why is there no mention of $\tilde f_1$?