Let $T = \mathbb{G}_m$ be the torus, and let $\tilde{T}$ be its étale universal cover (a pro-object in schemes of finite type). Then both $T$ and $\tilde{T}$ have a well-defined étale homotopy type. Explicitly, the homotopy types are $ét(T) = B\hat{\mathbb{Z}}$ and $ét(\tilde{T}) = *,$ for $*$ the point. In particular, the natural covering map $$\pi:\tilde{T}\to T$$ gives a basepoint (in a suitable homotopy sense) $$ét(\pi)$$ of $ét(T)$. Now group structure on $T$ lets us define a new point $\pi^2 := \pi*\pi,$ which is the composition of $\pi$ with the squaring map $[2]:T\to T$. While both $\pi, \pi^2:\tilde{T}\to T$ realize $\tilde{T}$ as a universal cover of $T$, they are not equal (as can be seen e.g. by looking at the map on tangent spaces at $1$). On the other hand, since $B\hat{\mathbb{Z}}$ is (I think?) connected, the maps $$ét(\pi), ét(\pi^2):ét(\tilde{T})\to ét(T)$$ should be homotopy equivalent in a suitable homotopy category.

**Question** is there a way to see the homotopy equivalence between $ét(\pi)$ and $ét(\pi^2)$ *explicitly*? Here by "explicitly", I mean as an interval in the mapping space between natural topological models, or an interval in some other model category. Edit: I'd also like for the functor from varieties to etale types to take etale maps to fibrations, so that in particular the etale types of $\pi, \pi^2$ are not a priori equal.