Consider the punctored line $X=\Bbb{A}^1_k\setminus \{s_1,\ldots,s_n\}$ over some field $k$. A(n étale) path in $X$ between two geometric points $x$ and $y$ is, by definition, an isomorphism between the fiber functors $\operatorname{Fib}_x$ and $\operatorname{Fib}_y$.
Here the functor $\operatorname{Fib}_x:\mathbf{FÉt}(X)\rightarrow \mathbf{Set}$ is the functor sending a finite étale cover $E$ of $X$ to the set underling the fiber $E_x$.
A path between $x$ and $y$ induce an isomorphism $\pi^{ét}_1(X,x)\simeq \pi^{ét}_1(X,y)$.
The spectrum of $\mathcal{O}^{sh}_{X,x}$ ($sh$=strict henselization) is a small (étale) neighborhood around $x$ and has essentially two geometric points, $x$ and the (separable closure of the) generic point $\eta$.
I'm wondering if there is a canonical way of choosing a path between $x$ and $\eta$, i.e. I'm wondering if there is a canonical way of choosing an isomorphism $$\pi_1^{ét}(X,x)\simeq \pi^{ét}_1(X,\eta)$$
