Theorem 15.2 (uniqueness of lifts):

Suppose $p: {\tilde X}\longrightarrow X$ is a covering projection, $T$ is a connected topological space and $f_1: T \longrightarrow {\tilde X}$ and $f_2: T \longrightarrow {\tilde X}$ are two lifts of a given continuous map $f: T \longrightarrow X$ such that $f_1(t_0) = f_2(t_0)$ for some $t_0\in T$. Then the two lifts agree on $T$ namely, $f_1(t) = f_2(t)$ for all $t \in T$.