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.