Let $f : X \to Y$ be a morphism of schemes. The functor $Y_{pro\text{-}\acute{e}tale}\to X_{pro\text{-}\acute{e}tale}$, $V \mapsto X \times _ Y V$ induces a morphism of sites $f_{pro\text{-}\acute{e}tale}: X_{pro\text{-}\acute{e}tale}\to Y_{pro\text{-}\acute{e}tale}$, see Sites, Proposition 7.14.7. In fact, we obtain a commutative diagram of morphisms of sites
where $\epsilon $ is as in Section 61.19. In particular we have $\epsilon ^{-1} f_{\acute{e}tale}^{-1} = f_{pro\text{-}\acute{e}tale}^{-1} \epsilon ^{-1}$. Here is the corresponding result for pushforward.
Lemma 61.23.1. Let $f : X \to Y$ be a morphism of schemes.
Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$. Then we have $f_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1}\mathcal{F} = \epsilon ^{-1}f_{{\acute{e}tale}, *}\mathcal{F}$.
Let $\mathcal{F}$ be an abelian sheaf on $X_{\acute{e}tale}$. Then we have $Rf_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1}\mathcal{F} = \epsilon ^{-1}Rf_{{\acute{e}tale}, *}\mathcal{F}$.
Proof. Proof of (1). Let $\mathcal{F}$ be a sheaf of sets on $X_{\acute{e}tale}$. There is a canonical map $\epsilon ^{-1}f_{{\acute{e}tale}, *}\mathcal{F} \to f_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1}\mathcal{F}$, see Sites, Section 7.45. To show it is an isomorphism we may work (Zariski) locally on $Y$, hence we may assume $Y$ is affine. In this case every object of $Y_{pro\text{-}\acute{e}tale}$ has a covering by objects $V = \mathop{\mathrm{lim}}\nolimits V_ i$ which are limits of affine schemes $V_ i$ étale over $Y$ (by Proposition 61.9.1 for example). Evaluating the map $\epsilon ^{-1}f_{{\acute{e}tale}, *}\mathcal{F} \to f_{{pro\text{-}\acute{e}tale}, *}\epsilon ^{-1}\mathcal{F}$ on $V$ we obtain a map
see Lemma 61.19.3 for the left hand side. By Lemma 61.19.3 we have
Hence the result holds by Étale Cohomology, Lemma 59.51.5.
Proof of (2). Arguing in exactly the same manner as above we see that it suffices to show that
which follows once more from Étale Cohomology, Lemma 59.51.5. $\square$
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