Lemma 76.9.2. Let S be a scheme. Let B be an algebraic space over S. Let X \subset X' and Y \subset Y' be thickenings of algebraic spaces over B. Let f : X \to Y be a morphism of algebraic spaces over B. Given any map of \mathcal{O}_ B-algebras
\alpha : f_{spaces, {\acute{e}tale}}^{-1}\mathcal{O}_{Y'} \to \mathcal{O}_{X'}
such that
\xymatrix{ f_{spaces, {\acute{e}tale}}^{-1}\mathcal{O}_ Y \ar[r]_-{f^\sharp } \ar[r] & \mathcal{O}_ X \\ f_{spaces, {\acute{e}tale}}^{-1}\mathcal{O}_{Y'} \ar[r]^-\alpha \ar[u]^{i_ Y^\sharp } & \mathcal{O}_{X'} \ar[u]_{i_ X^\sharp } }
commutes, there exists a unique morphism of (f, f') of thickenings over B such that \alpha = (f')^\sharp .
Proof.
To find f', by Properties of Spaces, Theorem 66.28.4, all we have to do is show that the morphism of ringed topoi
(f_{spaces, {\acute{e}tale}}, \alpha ) : (\mathop{\mathit{Sh}}\nolimits (X_{spaces, {\acute{e}tale}}), \mathcal{O}_{X'}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (Y_{spaces, {\acute{e}tale}}), \mathcal{O}_{Y'})
is a morphism of locally ringed topoi. This follows directly from the definition of morphisms of locally ringed topoi (Modules on Sites, Definition 18.40.9), the fact that (f, f^\sharp ) is a morphism of locally ringed topoi (Properties of Spaces, Lemma 66.28.1), that \alpha fits into the given commutative diagram, and the fact that the kernels of i_ X^\sharp and i_ Y^\sharp are locally nilpotent. Finally, the fact that f' \circ i_ X = i_ Y \circ f follows from the commutativity of the diagram and another application of Properties of Spaces, Theorem 66.28.4. We omit the verification that f' is a morphism over B.
\square
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