Lemma 81.12.5. Let $(R \to R', f)$ be a glueing pair, see above. The functor (81.12.0.1) is fully faithful on the full subcategory of algebraic spaces $Y/X$ which are (a) glueable for $(R \to R', f)$ and (b) have affine diagonal $Y \to Y \times _ X Y$.
Proof. Let $Y, Z$ be two algebraic spaces over $X$ which are both glueable for $(R \to R', f)$ and assume the diagonal of $Z$ is affine. Let $a : U \times _ X Y \to U \times _ X Z$ over $U$ and $b : X' \times _ X Y \to X' \times _ X Z$ over $X'$ be two morphisms of algebraic spaces which induce the same morphism $c : U' \times _ X Y \to U' \times _ X Z$ over $U'$. We want to construct a morphism $f : Y \to Z$ over $X$ which produces the morphisms $a$, $b$ on base change to $U$, $X'$. By the faithfulness of Lemma 81.12.4, it suffices to construct the morphism $f$ étale locally on $Y$ (details omitted). Thus we may and do assume $Y$ is affine.
Let $y \in |Y|$ be a point. If $y$ maps into the open $U \subset X$, then $U \times _ X Y$ is an open of $Y$ on which the morphism $f$ is defined (we can just take $a$). Thus we may assume $y$ maps into the closed subset $V(f)$ of $X$. Since $R/fR = R'/fR'$ there is a unique point $y' \in |X' \times _ X Y|$ mapping to $y$. Denote $z' = b(y') \in |X' \times _ X Z|$ and $z \in |Z|$ the images of $y'$. Choose an étale neighbourhood $(W, w) \to (Z, z)$ with $W$ affine. Observe that
\[ (U \times _ X W) \times _{U \times _ X Z, a} (U \times _ X Y),\quad (U' \times _ X W) \times _{U' \times _ X Z, c} (U' \times _ X Y), \]
and
\[ (X' \times _ X W) \times _{X' \times _ X Z, b} (X' \times _ X Y) \]
form an object of $\textit{Spaces}(U \leftarrow U' \to X')$ with affine parts (this is where we use that $Z$ has affine diagonal). Hence by Lemma 81.12.2 there exists a unique affine scheme $V$ glueable for $(R \to R', f)$ such that
\[ (U \times _ X V, U' \times _ X V, X' \times _ X V) \]
is the triple displayed above. By fully faithfulness for the affine case (Lemma 81.12.2) we get a unique morphisms $V \to W$ and $V \to Y$ agreeing with the first and second projection morphisms over $U$ and $X'$ in the construction above. By Lemma 81.12.3 the morphism $V \to Y$ is étale. To finish the proof, it suffices to show that there is a point $v \in |V|$ mapping to $y$ (because then $f$ is defined on an étale neighbourhood of $y$, namely $V$). There is a unique point $w' \in |X' \times _ X W|$ mapping to $w$. By uniqueness $w'$ is mapped to $z'$ under the map $|X' \times _ X W| \to |X' \times _ X Z|$. Then we consider the cartesian diagram
\[ \xymatrix{ X' \times _ X V \ar[r] \ar[d] & X' \times _ X W \ar[d] \\ X' \times _ X Y \ar[r] & X' \times _ X Z } \]
to see that there is a point $v' \in |X' \times _ X V|$ mapping to $y'$ and $w'$, see Properties of Spaces, Lemma 66.4.3. Of course the image $v$ of $v'$ in $|V|$ maps to $y$ and the proof is complete. $\square$
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