ag.algebraic geometry - Does an étale equivalence relation of schemes induce an equivalence relation on points?

Let $R rightrightarrows U to X$ be a presentation of an algebraic space by schemes.
Does this induce an exact sequence $|R| rightrightarrows |U| to |X|$ on underlying points?

The reason I ask is that this is stated as a lemma in the Stack Project (Lemma 33.4.4). However, the proof uses:

"Since $j = (s,t):R to U times U$ is a monomorphism we see that $|R| to |U|times|U|$ is injective."

But this can't be true in general. For instance $|U times U| to |U|times|U|$ is not injective when $U$ is the affine line since all generic points in $|Utimes U|$ corresponding to dimension 1 curves map to the same pair of generic points. Of course, this equivalence relation is not étale. The question is: Does étaleness prevent such collapses to happen? In that case, how?

Edit: It is pretty clear that $|R| rightrightarrows |U|$
becomes a pre-equivalence relation with $|U| to |X|$ as quotient.
In fact, I think this is the only property used later in the text, which makes the error unproblematic.

Then the only question is: Is $|R| to |U| times_{|X|}|U|$ injective?