The Stacks project

Lemma 7.32.9. Let $\mathcal{C}$ be a site. Let $p : \mathop{\mathit{Sh}}\nolimits (pt) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C})$ be a point of the topos associated to $\mathcal{C}$. For any set $E$ there are canonical maps

\[ E \longrightarrow (p_*E)_ p \longrightarrow E \]

whose composition is $\text{id}_ E$.

Proof. There is always an adjunction map $(p_*E)_ p = p^{-1}p_*E \to E$. This map is an isomorphism when $E = \{ *\} $ because $p_*$ and $p^{-1}$ are both left exact, hence transform the final object into the final object. Hence given $e \in E$ we can consider the map $i_ e : \{ *\} \to E$ which gives

\[ \xymatrix{ p^{-1}p_*\{ *\} \ar[rr]_{p^{-1}p_*i_ e} \ar[d]_{\cong } & & p^{-1}p_*E \ar[d] \\ \{ *\} \ar[rr]^{i_ e} & & E } \]

whence the map $E \to (p_*E)_ p = p^{-1}p_*E$ as desired. $\square$

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