There is a theorem of Cartan which states that if $M$ is a simply connected, complete Riemann manifold, and that the sectional curvature is everywhere $\leq 0$, then any two points of M are joined by a unique geodesic.
So if we consider a Riemann manifold $M$ which is not simply connected, but is complete and has sectional curvature $\leq 0$, we could use the above theorem of Cartan to the universal covering space $\widetilde{M}$ of $M$. It says in the book:
For it is clear that $\widetilde{M}$ inherits a Riemannian metric from $M$ which is geodesically complete, and has sectioanl curvature $\leq 0$.
Given two points $p,q \in M$, it follows that each homotopy class of paths from p to q contains precisely one geodesic.
My question is: How does the second sectence deduced from the sentence above? I know that any two points of $\widetilde{M}$ are joined by only one geodesic, but for any two points $p,q \in M$, there are many lifted points of $p,q$ in $\widetilde{M}$. Thank you!