Assume that $f:X\to Y$ is an étale map between smooth varieties and $(S,I)$ is a Henselian pair. Let $\alpha\in X(S/I)$. Can we say that the lifts of $\alpha$ to $X(S)$ are in bijection with the lifts of $f(\alpha)$ to $Y(S)$? Can we at least say that there is an injection?
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I think that the answer is that there is a bijection between lifts to $Y(S)$ and lifts to $X(S)$. This is probably well-known, and can be obtained by reduction to the affine case (which readily available in the literature).
Since a localization of $S$ with respect to an element $f$ can be written as a filtered colimit $$S \xrightarrow{f} S \xrightarrow{f} S ...$$ it follows from https://stacks.math.columbia.edu/tag/0FWT that the localization $(S_f, I_f)$ is a Henselian pair. By Zariski gluing, we may choose finitely many elements $f_1, .., f_n \in S$ such that $Spec(S_{f_i})$ cover of $Spec(S)$ and work with $S_{f_i}$ instead. Moreover, after refining the cover, we may choose $f_i$ such that the morphisms $Spec(S_{f_i}/I_{f_i}) \to X \to Y$ factor through affine opens $X_i \subset X$ and $Y_i \subset Y$. Since every point of $Spec(S_{f_i})$ specializes to a point of $Spec(S_{f_i}/I_{f_i})$ (because $I$ is in the Jacobson radical), any lifts to $Spec(S_{f_i})$ will also factor through the same affine opens $X_i, Y_i$. Therefore, replacing $S, X, Y$ with $S_{f_i}, X_i, Y_i$, we may assume that everything is affine.
Now choose a morphism $Spec(S) \to Y$ lifting the original $Spec(S/I) \to Y$. We want to show that there is a unique lift to $X(S)$ that restricts to the original point in $X(S/I)$. After base-changing to $Spec(S)$, we may assume that $Y = Spec(S)$. The existence now is one of the equivalent conditions of a Henselian pair (see for example Gabber's paper "Affine analogs of the proper base change theorem" part (c) in the introduction).
