Lemma 66.27.1. Let $S$ be a scheme. Let

\[ \xymatrix{ U \ar[d]_ p \ar[r]_ g & V \ar[d]^ q \\ X \ar[r]^ f & Y } \]

be a commutative diagram of algebraic spaces over $S$ with $p$ and $q$ étale. Via the identifications (66.27.0.2) for $U \to X$ and $V \to Y$ the morphism of ringed topoi

\[ (g_{spaces, {\acute{e}tale}}, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (U_{spaces, {\acute{e}tale}}), \mathcal{O}_ U) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (V_{spaces, {\acute{e}tale}}), \mathcal{O}_ V) \]

is $2$-isomorphic to the morphism $(f_{spaces, {\acute{e}tale}, c}, f_ c^\sharp )$ constructed in Modules on Sites, Lemma 18.20.2 starting with the morphism of ringed sites $(f_{spaces, {\acute{e}tale}}, f^\sharp )$ and the map $c : U \to V \times _ Y X$ corresponding to $g$.

**Proof.**
The morphism $(f_{spaces, {\acute{e}tale}, c}, f_ c^\sharp )$ is defined as a composition $f' \circ j$ of a localization and a base change map. Similarly $g$ is a composition $U \to V \times _ Y X \to V$. Hence it suffices to prove the lemma in the following two cases: (1) $f = \text{id}$, and (2) $U = X \times _ Y V$. In case (1) the morphism $g : U \to V$ is étale, see Lemma 66.16.6. Hence $(g_{spaces, {\acute{e}tale}}, g^\sharp )$ is a localization morphism by the discussion surrounding Equations (66.27.0.1) and (66.27.0.2) which is exactly the content of the lemma in this case. In case (2) the morphism $g_{spaces, {\acute{e}tale}}$ comes from the morphism of ringed sites given by the functor $V_{spaces, {\acute{e}tale}} \to U_{spaces, {\acute{e}tale}}$, $V'/V \mapsto V' \times _ V U/U$ which is also what the morphism $f'$ is defined by, see Sites, Lemma 7.28.1. We omit the verification that $(f')^\sharp = g^\sharp $ in this case (both are the restriction of $f^\sharp $ to $U_{spaces, {\acute{e}tale}}$).
$\square$

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