Clarifying a lemma

This is much better because for example now we can see that
the Y that is produced by the lemma is unique, satisfies a
universal property, etc, etc.

Thanks to comment_bot
https://stacks.math.columbia.edu/tag/0F9U#comment-4313

spaces-pushouts.tex

@@ -1831,8 +1831,9 @@ \section{Glueing and the Beauville-Laszlo theorem}
 \label{lemma-glueing-quasi-affines}
 Let $(R \to R', f)$ be a glueing pair, see above. Any object
 $(V, V', Y')$ of $\textit{Spaces}(U \leftarrow U' \to X')$
-with $V$, $V'$, $Y'$ quasi-affine is in the essential image
-of the functor (\ref{equation-beauville-laszlo-glueing-spaces}).
+with $V$, $V'$, $Y'$ quasi-affine is isomorphic to
+the image under the functor (\ref{equation-beauville-laszlo-glueing-spaces})
+of a separated algebraic space $Y$ over $X$.
 \end{lemma}
 
 \begin{proof}
@@ -1906,8 +1907,13 @@ \section{Glueing and the Beauville-Laszlo theorem}
 $$
 is an algebraic space for example by
 Bootstrap, Theorem \ref{bootstrap-theorem-final-bootstrap}.
-Since it is clear that $Y/X$ is sent to the triple $(V, V', Y')$
-the proof is complete.
+Since it is clear that $Y/X$ is sent to the triple $(V, V', Y')$.
+The base change of the diagonal $\Delta : Y \to Y \times_X Y$
+by the quasi-compact surjective flat morphism $T \times_X T \to Y \times_X Y$
+is the closed immersion $W \to T \times_X T$. Thus $\Delta$
+is a closed immersion by Descent on Spaces, Lemma
+\ref{spaces-descent-lemma-descending-property-closed-immersion}.
+Thus the algebraic space $Y$ is separated and the proof is complete.
 \end{proof}