ag.algebraic geometry - Do quotients of representable sheaves represent quotients?

Here's the context for the question: Proposition 4.6 of Freitag and Kiehl's book on etale cohomology shows that a sheaf (of sets) $mathcal{F}$ (on the site Et(X)) is constructible if and only if it is the coequalizer of an etale equivalence relation $mathcal{R}rightrightarrows mathcal{Y}$, where $mathcal{R}$ and $mathcal{Y}$ are representable sheaves. Here an etale equivalence relation is defined exactly as you would expect: $mathcal{R}rightarrow mathcal{Y}times mathcal{Y}$ is injective, and for every etale $Urightarrow X$, $mathcal{R}(U)subset mathcal{Y}(U)times mathcal{Y}(U)$ is an equivalence relation (of sets).

Now if the quotient $mathcal{Y}/mathcal{R}$ "should" be represented by the quotient $Y/R$ (where $Y$ represents $mathcal{Y}$ and $R$ represents $mathcal{R}$), well, it sounds like constructible sheaves should be algebraic spaces, or at least there should be some relationship. On the other hand, I don't think this could be right.

So is the problem that you can't take sheafy quotients like this, or is it something more subtle?