Remark 73.4.14. The sites $(\textit{Spaces}/X)_{\acute{e}tale}$ and $X_{spaces, {\acute{e}tale}}$ come with structure sheaves. For the small étale site we have seen this in Properties of Spaces, Section 66.21. The structure sheaf $\mathcal{O}$ on the big étale site $(\textit{Spaces}/X)_{\acute{e}tale}$ is defined by assigning to an object $U$ the global sections of the structure sheaf of $U$. This makes sense because after all $U$ is an algebraic space itself hence has a structure sheaf. Since $\mathcal{O}_ U$ is a sheaf on the étale site of $U$, the presheaf $\mathcal{O}$ so defined satisfies the sheaf condition for coverings of $U$, i.e., $\mathcal{O}$ is a sheaf. We can upgrade the morphisms $i_ f$, $\pi _ X$, $i_ X$, $f_{small}$, and $f_{big}$ defined above to morphisms of ringed sites, respectively topoi. Let us deal with these one by one.

In Lemma 73.4.7 denote $\mathcal{O}$ the structure sheaf on $(\textit{Spaces}/X)_{\acute{e}tale}$. We have $(i_ f^{-1}\mathcal{O})(U/Y) = \mathcal{O}_ U(U) = \mathcal{O}_ Y(U)$ by construction. Hence an isomorphism $i_ f^\sharp : i_ f^{-1}\mathcal{O} \to \mathcal{O}_ Y$.

In Lemma 73.4.8 it was noted that $i_ X$ is a special case of $i_ f$ with $f = \text{id}_ X$ hence we are back in case (1).

In Lemma 73.4.8 the morphism $\pi _ X$ satisfies $(\pi _{X, *}\mathcal{O})(U) = \mathcal{O}(U) = \mathcal{O}_ X(U)$. Hence we can use this to define $\pi _ X^\sharp : \mathcal{O}_ X \to \pi _{X, *}\mathcal{O}$.

In Lemma 73.4.11 the extension of $f_{small}$ to a morphism of ringed topoi was discussed in Properties of Spaces, Lemma 66.21.3.

In Lemma 73.4.11 the functor $f_{big}^{-1}$ is simply the restriction via the inclusion functor $(\textit{Spaces}/Y)_{\acute{e}tale}\to (\textit{Spaces}/X)_{\acute{e}tale}$. Let $\mathcal{O}_1$ be the structure sheaf on $(\textit{Spaces}/X)_{\acute{e}tale}$ and let $\mathcal{O}_2$ be the structure sheaf on $(\textit{Spaces}/Y)_{\acute{e}tale}$. We obtain a canonical isomorphism $f_{big}^\sharp : f_{big}^{-1}\mathcal{O}_1 \to \mathcal{O}_2$.

Moreover, with these definitions compositions work out correctly too. We omit giving a detailed statement and proof.

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