Lemma 84.4.4. Let S be a scheme. Let f : X \to Y and g : Y' \to Y be a morphisms of algebraic spaces over S. Assume f is proper. Set X' = Y' \times _ Y X with projections f' : X' \to Y' and g' : X' \to X. Let \mathcal{F} be any sheaf on X_{\acute{e}tale}. Then g^{-1}f_*\mathcal{F} = f'_*(g')^{-1}\mathcal{F}.
Proof. The question is étale local on Y'. Choose a scheme V and a surjective étale morphism V \to Y. Choose a scheme V' and a surjective étale morphism V' \to V \times _ Y Y'. Then we may replace Y' by V' and Y by V. Hence we may assume Y and Y' are schemes. Then we may work Zariski locally on Y and Y' and hence we may assume Y and Y' are affine schemes.
Assume Y and Y' are affine schemes. Choose a surjective proper morphism h_1 : X_1 \to X where X_1 is a scheme, see Cohomology of Spaces, Lemma 69.18.1. Set X_2 = X_1 \times _ X X_1 and denote h_2 : X_2 \to X the structure morphism. Observe this is a scheme. By the case of schemes (Étale Cohomology, Lemma 59.91.5) we know the lemma is true for the cartesian diagrams
\vcenter { \xymatrix{ X'_1 \ar[r] \ar[d] & X_1 \ar[d] \\ Y' \ar[r] & Y } } \quad \text{and}\quad \vcenter { \xymatrix{ X'_2 \ar[r] \ar[d] & X_2 \ar[d] \\ Y' \ar[r] & Y } }
and the sheaves \mathcal{F}_ i = (X_ i \to X)^{-1}\mathcal{F}. By Lemma 84.4.1 we have an exact sequence 0 \to \mathcal{F} \to h_{1, *}\mathcal{F}_1 \to h_{2, *}\mathcal{F}_2 and similarly for (g')^{-1}\mathcal{F} because X'_2 = X'_1 \times _{X'} X'_1. Hence we conlude that the lemma is true (some details omitted). \square
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