The Stacks project

Proof. The morphism $(f_{spaces, {\acute{e}tale}, c}, f_ c^\sharp )$ is defined as a composition $f' \circ j$ of a localization and a base change map. Similarly $g$ is a composition $U \to V \times _ Y X \to V$. Hence it suffices to prove the lemma in the following two cases: (1) $f = \text{id}$, and (2) $U = X \times _ Y V$. In case (1) the morphism $g : U \to V$ is étale, see Lemma 66.16.6. Hence $(g_{spaces, {\acute{e}tale}}, g^\sharp )$ is a localization morphism by the discussion surrounding Equations (66.27.0.1) and (66.27.0.2) which is exactly the content of the lemma in this case. In case (2) the morphism $g_{spaces, {\acute{e}tale}}$ comes from the morphism of ringed sites given by the functor $V_{spaces, {\acute{e}tale}} \to U_{spaces, {\acute{e}tale}}$, $V'/V \mapsto V' \times _ V U/U$ which is also what the morphism $f'$ is defined by, see Sites, Lemma 7.28.1. We omit the verification that $(f')^\sharp = g^\sharp $ in this case (both are the restriction of $f^\sharp $ to $U_{spaces, {\acute{e}tale}}$). $\square$

Proof. Note that $(g_{small}, g^\sharp )$ is $2$-isomorphic as a morphism of ringed topoi to the morphism of ringed topoi associated to the morphism of ringed sites $(g_{spaces, {\acute{e}tale}}, g^\sharp )$. Hence we conclude by Lemma 66.27.1 and Modules on Sites, Lemma 18.22.4. $\square$

Proof. Let us prove that $X$ is a sheaf for the fppf topology. Namely, suppose that $\{ g_ i : T_ i \to T\} $ is a covering of $(\mathit{Sch}/S)_{fppf}$ and $(y_ i, s_ i) \in X(T_ i)$ satisfy the glueing condition, i.e., the restriction of $(y_ i, s_ i)$ and $(y_ j, s_ j)$ to $T_ i \times _ T T_ j$ agree. Then since $Y$ is a sheaf for the fppf topology, we see that the $y_ i$ give rise to a unique morphism $y : T \to Y$ such that $y_ i = y \circ g_ i$. Then we see that $y_{i, small}^{-1}\mathcal{F} = g_{i, small}^{-1}y_{small}^{-1}\mathcal{F}$. Hence the sections $s_ i$ glue uniquely to a section of $y_{small}^{-1}\mathcal{F}$ by Étale Cohomology, Lemma 59.39.2.

The construction that sends $\mathcal{F} \in \mathop{\mathrm{Ob}}\nolimits (\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale}))$ to $X \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ preserves finite limits and all colimits since each of the functors $y_{small}^{-1}$ have this property. Of course, if $V \in \mathop{\mathrm{Ob}}\nolimits (Y_{\acute{e}tale})$, then the construction sends the representable sheaf $h_ V$ on $Y_{\acute{e}tale}$ to the representable functor represented by $V$.

By Sites, Lemma 7.12.5 we can find a set $I$, for each $i \in I$ an object $V_ i$ of $Y_{\acute{e}tale}$ and a surjective map of sheaves

on $Y_{\acute{e}tale}$. The set theoretic condition we need is that the index set $I$ is not too large¹. Then $V = \coprod V_ i$ is an object of $(\mathit{Sch}/S)_{fppf}$ and therefore an object of $Y_{\acute{e}tale}$ and we have a surjective map $h_ V \to \mathcal{F}$.

Observe that the product of $h_ V$ with itself in $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$ is $h_{V \times _ Y V}$. Consider the fibre product

There is an open subscheme $R$ of $V \times _ Y V$ such that $h_ V \times _\mathcal {F} h_ V = h_ R$, see Lemma 66.20.1 (small detail omitted). By the Yoneda lemma we obtain two morphisms $s, t : R \to V$ in $Y_{\acute{e}tale}$ and we find a coequalizer diagram

in $\mathop{\mathit{Sh}}\nolimits (Y_{\acute{e}tale})$. Of course the morphisms $s, t$ are étale and define an étale equivalence relation $(t, s) : R \to V \times _ S V$.

By the discussion in the preceding two paragraphs we find a coequalizer diagram

in $(\mathit{Sch}/S)_{fppf}$. Thus $X = V/R$ is an algebraic space by Spaces, Theorem 65.10.5. The other statements follow readily from this; details omitted. $\square$