= Solution
Let $p:\widehat X\to X$ be the covering map. The <uniqueness of lifts to a covering space> says that if $\widehat f_1,\widehat f_2:Y\to\widehat X$ satisfy
$$
p\circ\widehat f_1=f=p\circ\widehat f_2
$$
and $\widehat f_1(y_0)=\widehat f_2(y_0)$ at one point $y_0\in Y$, then
$$
\boxed{\widehat f_1=\widehat f_2}.
$$
For any $y\in Y$, choose a path $\gamma$ from $y_0$ to $y$, using path-connectedness of $Y$. Both $\widehat f_1\circ\gamma$ and $\widehat f_2\circ\gamma$ lift the same path $f\circ\gamma$ and have the same initial point. Uniqueness in the <path lifting theorem> makes the lifted paths equal, so their endpoints agree. Since $y$ was arbitrary, the two lifts agree everywhere.
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