Let $M$ be a connected manifold with universal cover $\tilde M$ and fix $x_0 \in M$. Then it is well-known that $\tilde M \to M$ is a principal $\pi_1(M,x_0)$ bundle. I'm a bit confused about the right action $\tilde M \times \pi_1(M,x_0) \to \tilde M$. The fiber over $x \in M$ is naturally $\pi_1(M,x)$ which, by choosing a path between $x$ and $x_0$, is isomorphic to $\pi_1(M,x_0)$ and so the right action on the fiber is just given by multiplication in the group $\pi_1(M)$. But for this action to be continuous it seems that the paths between $x$ and $x_0$ have to be chosen in a continuous way.
Is this the right way to think about this? If so, what's a clean way of seeing how to choose the paths?