Lemma 63.4.3. Let $f = j : U \to X$ be an étale of schemes. Denote $j_{p!}$ the construction of Étale Cohomology, Equation (59.70.1.1) and denote $f_{p!}$ the construction above. Functorially in $\mathcal{F} \in \textit{Ab}(X_{\acute{e}tale})$ there is a canonical map
\[ j_{p!}\mathcal{F} \longrightarrow f_{p!}\mathcal{F} \]
of abelian presheaves which identifies the sheaf $j_!\mathcal{F} = (j_{p!}\mathcal{F})^\# $ of Étale Cohomology, Definition 59.70.1 with $(f_{p!}\mathcal{F})^\# $.
Proof.
Please read the proof of Étale Cohomology, Lemma 59.70.6 before reading the proof of this lemma. Let $V$ be an object of $X_{\acute{e}tale}$. Recall that
\[ j_{p!}\mathcal{F}(V) = \bigoplus \nolimits _{\varphi : V \to U} \mathcal{F}(V \xrightarrow {\varphi } U) \]
Given $\varphi $ we obtain an open subscheme $Z_\varphi \subset U_ V = U \times _ X V$, namely, the image of the graph of $\varphi $. Via $\varphi $ we obtain an isomorphism $V \to Z_\varphi $ over $U$ and we can think of an element
\[ s_\varphi \in \mathcal{F}(V \xrightarrow {\varphi } U) = \mathcal{F}(Z_\varphi ) = H_{Z_\varphi }(\mathcal{F}) \]
as a section of $\mathcal{F}$ over $Z_{\varphi }$. Since $Z_\varphi \subset U_ V$ is open, we actually have $H_{Z_\varphi }(\mathcal{F}) = \mathcal{F}(Z_\varphi )$ and we can think of $s_\varphi $ as an element of $H_{Z_\varphi }(\mathcal{F})$. Having said this, our map $j_{p!}\mathcal{F} \to f_{p!}\mathcal{F}$ is defined by the rule
\[ \sum \nolimits _{i = 1, \ldots , n} s_{\varphi _ i} \longmapsto \sum \nolimits _{i = 1, \ldots , n} (Z_{\varphi _ i}, s_{\varphi _ i}) \]
with right hand side a sum as in (63.4.0.1). We omit the verification that this is compatible with restriction mappings and functorial in $\mathcal{F}$.
To finish the proof, we claim that given a geometric point $\overline{y} : \mathop{\mathrm{Spec}}(k) \to Y$ there is a commutative diagram
\[ \xymatrix{ (j_{p!}\mathcal{F})_{\overline{y}} \ar[r] \ar[d] & \bigoplus _{j(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} \ar@{=}[d] \\ (f_{p!}\mathcal{F})_{\overline{y}} \ar[r] & \bigoplus _{f(\overline{x}) = \overline{y}} \mathcal{F}_{\overline{x}} } \]
where the top horizontal arrow is constructed in the proof of Étale Cohomology, Proposition 59.70.3, the bottom horizontal arrow is constructed in the proof of Lemma 63.4.2, the right vertical arrow is the obvious equality, and the left vertical arrow is the map defined in the previous paragraph on stalks. The claim follows in a straightforward manner from the explicit description of all of the arrows involved here and in the references given. Since the horizontal arrows are isomorphisms we conclude so is the left vertical arrow. Hence we find that our map induces an isomorphism on sheafifications by Étale Cohomology, Theorem 59.29.10.
$\square$
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