Uniqueness of lifts to a covering space
= Uniqueness of lifts to a covering space
Let $p:\widetilde X\to X$ be a covering map and let $Y$ be path-connected. If two lifts $\widetilde f_1,\widetilde f_2:Y\to\widetilde X$ of the same map $f:Y\to X$ agree at one point, they agree everywhere. Join that point to any other by a path and apply uniqueness in the <path lifting theorem>.