Lemma 69.6.1. Let $S$ be a scheme. Let $f_ i : U_ i \to X$ be étale morphisms of algebraic spaces over $S$. Then there are isomorphisms
\[ f_{1, !}\underline{\mathbf{Z}} \otimes _{\mathbf{Z}} f_{2, !}\underline{\mathbf{Z}} \longrightarrow f_{12, !}\underline{\mathbf{Z}} \]
where $f_{12} : U_1 \times _ X U_2 \to X$ is the structure morphism and
\[ (f_1 \amalg f_2)_! \underline{\mathbf{Z}} \longrightarrow f_{1, !}\underline{\mathbf{Z}} \oplus f_{2, !}\underline{\mathbf{Z}} \]
Proof.
Once we have defined the map it will be an isomorphism by our description of stalks above. To define the map it suffices to work on the level of presheaves. Thus we have to define a map
\[ \left(\bigoplus \nolimits _{\varphi _1 \in \mathop{\mathrm{Mor}}\nolimits _ X(V, U_1)} \mathbf{Z}\right) \otimes _{\mathbf{Z}} \left(\bigoplus \nolimits _{\varphi _2 \in \mathop{\mathrm{Mor}}\nolimits _ X(V, U_2)} \mathbf{Z}\right) \longrightarrow \bigoplus \nolimits _{\varphi \in \mathop{\mathrm{Mor}}\nolimits _ X(V, U_1 \times _ X U_2)} \mathbf{Z} \]
We map the element $1_{\varphi _1} \otimes 1_{\varphi _2}$ to the element $1_{\varphi _1 \times \varphi _2}$ with obvious notation. We omit the proof of the second equality.
$\square$
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