Lemma 59.46.3. Let $i : Z \to X$ be a closed immersion of schemes. Let $\mathcal{G}$ be a sheaf of sets on $Z_{\acute{e}tale}$. Let $\overline{x}$ be a geometric point of $X$. Then
where $*$ denotes a singleton set.
Proof. Note that $i_{small, *}\mathcal{G}|_{U_{\acute{e}tale}} = *$ is the final object in the category of étale sheaves on $U$, i.e., the sheaf which associates a singleton set to each scheme étale over $U$. This explains the value of $(i_{small, *}\mathcal{G})_{\overline{x}}$ if $\overline{x} \not\in Z$.
Next, suppose that $\overline{x} \in Z$. Note that
and on the other hand
Let $\mathcal{C}_1 = \{ (U, \overline{u})\} ^{opp}$ be the opposite of the category of étale neighbourhoods of $\overline{x}$ in $X$, and let $\mathcal{C}_2 = \{ (V, \overline{v})\} ^{opp}$ be the opposite of the category of étale neighbourhoods of $\overline{x}$ in $Z$. The canonical map
corresponds to the functor $F : \mathcal{C}_1 \to \mathcal{C}_2$, $F(U, \overline{u}) = (U_ Z, \overline{x})$. Now Lemmas 59.46.2 and 59.46.1 imply that $\mathcal{C}_1$ is cofinal in $\mathcal{C}_2$, see Categories, Definition 4.17.1. Hence it follows that the displayed arrow is an isomorphism, see Categories, Lemma 4.17.2. $\square$
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