Let $j: U \hookrightarrow X$ be an open subscheme of an $S$-scheme $X$ and let $R \hookrightarrow X \times_S X$ be an etale equivalence relation on $X$. Denote by $R_u$ the fiber product of the diagram
$\require{AMScd}$ \begin{CD} R_U @>{}>> U \times_S U\\ @VVV @VVV\\ R @>{}>> X \times_S X. \end{CD}
Then $R_U$ is an equivalence relation on $U$, and the induced map $\overline{j}: U/R_U \to Y$, where $Y$ denotes the algebraic space $X/R$ is a monomorphism.
The statement of Lemma 5.2.8 is as follows:
The morphism $\overline{j}: U/R_U \to Y$ is representable by an open imbedding.
My Questions:
(1) Is it correct that the statement that $\overline{j}$ be representable by an open imbedding means that for each $S$-scheme $T$ and morphism $f: T \to Y$ the top horizontal arrow of the fiber product $\require{AMScd}$ \begin{CD} U/R_U \times_Y T @>{}>> T\\ @VVV @VVV\\ U/R_U @>{\overline{j}}>> Y. \end{CD} is an open imbedding?
(2) A morphism $f: T \to Y$ is contained in the T-points of $U/R_U$ if and only if etale locally on T the morphism $f$ factors through the open subset $U \subset X$. (This is in the proof of Lemma 5.2.8). Following this statement, Olsson then write that this condition on $f$ is representable by an open subset of $T$.
What does the statement "*this condition on $f$ is representable by an open subset of $T$" mean?
(1) You are right.
(2) Actually here $f$ is representable by an open subset of $T$ is not precise. We should say: $U/R_U\times_YT\to T$ is representable by an open subscheme of $T$.
The proof of this lemma in Olsson's book is actually a sketch. We should use some descent theory to make this open subset global. See the detailed proof of this Lemma in Stacks Project: https://stacks.math.columbia.edu/tag/02WU