**Proof.**
Parts (1) and (2) are equivalent by general properties of $1$-morphisms of categories fibred in groupoids, see Categories, Lemma 4.35.8. We see that (3) implies (2) by Lemma 92.9.2. Finally, assume (2). Let $U$ be a scheme. Let $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{Y}_ U)$. We have to prove that

\[ \mathcal{W} = (\mathit{Sch}/U)_{fppf} \times _{y, \mathcal{Y}} \mathcal{X} \]

is representable by an algebraic space over $U$. Since $(\mathit{Sch}/U)_{fppf}$ is an algebraic stack we see from Lemma 92.14.3 that $\mathcal{W}$ is an algebraic stack. On the other hand the explicit description of objects of $\mathcal{W}$ as triples $(V, x, \alpha : y(V) \to f(x))$ and the fact that $f$ is faithful, shows that the fibre categories of $\mathcal{W}$ are setoids. Hence Proposition 92.13.3 guarantees that $\mathcal{W}$ is representable by an algebraic space.
$\square$

## Comments (0)

There are also: