# etale homotopy and Adams conjecture

I was reading Quillen's paper on "Some remarks on etale homotopy theory and a conjecture of Adams". Up to some fact about etale version of spherical fibrations which was later proved by Friedlander, Quillen gives a proof of Adams conjecture using etale homotopy theory. Somewhere in middle of paper he is saying: Assume $Q$ is a vector bundle over variety $X$ in characteristic $p$. Assume that $Q^{(p)}$ is the pullback of $Q$ over $X$ by Frobenius map $\Phi : X\to X$ then there is a morphism $F: Q \rightarrow Q^{(p)}$ given by universal property. In local coordinates $F$ is given by $X_i\to X_i^p$. Note that it is not a bundle map but it preserves fibers and sends $Q-0$ to $Q^{(p)}-0$ so it induces a "homomorphism" ( I am not sure in what sense) in etale topology which is purely inseparable. Then Quillen concludes that $(Q-0)_{et} \simeq (Q^{(p)}-0) _{et}$. Can someone help me understand why these two have same etale homotopy type?

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