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If $f:X\rightarrow Y$ is a morphism of schemes, we get an induced morphism of (small) etale sites $X_{et}\rightarrow Y_{et}$ whose underlying functor is base change along $f$. Any commutative $Y$-group scheme $G$ defines an abelian sheaf $h_G=\mathrm{Hom}_{\mathrm{Sch}/Y}(-,G)$ on $Y_{et}$. I'm interested in knowing conditions where the pullback $f^{-1}(h_G)$ can be identified with $h_{G\times_YX}$ as sheaves on $X_{et}$. I know of two sufficient conditions:

(1) $G/Y$ is an object of $Y_{et}$ (in this case the desired equality follows from the adjunction between $f^{-1}$ and $f_*$ together with a couple applications of Yoneda's lemma)

(2) $X/Y$ is an object of $Y_{et}$ (I believe in this case the desired equality follows because the pullback $f^{-1}h_G$ is given by $(f^{-1}h_G)(U/X)=h_G(U/Y)=h_{G\times_YX}(U)$)

Are there any other general sufficient conditions? In general there is a morphism $f^{-1}h_G\rightarrow h_{G\times_YX}$. The way I think about this is in terms of the pullback-pushforward adjunction and the map $h_G\rightarrow f_*(h_{G\times_YX})$ given by the maps $G(V)\rightarrow(G\times_YX)(V\times_YX)$ coming from base change for $V/Y\in Y_{et}$.

I would be happy to know this map is an isomorphism when $G$ is smooth, or even more specifically, when $G$ is smooth, $X=\mathrm{Spec}(K)$, and $Y=\mathrm{Spec}(k)$ for an extension $K/k$ (not necessarily algebraic).

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