The Stacks project

Proof. Let $(V' \to Y', W \to Z, \alpha )$ be an object of the right hand side. Recall that $V'$, resp. $W$ gives rise to a representable sheaf $\mathcal{G}' = h_{V'}$ in $\mathop{\mathit{Sh}}\nolimits (Y'_{spaces, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (Y'_{\acute{e}tale})$, resp. $\mathcal{G} = h_ W$ in $\mathop{\mathit{Sh}}\nolimits (Z_{spaces, {\acute{e}tale}}) = \mathop{\mathit{Sh}}\nolimits (Z_{\acute{e}tale})$, see Properties of Spaces, Section 66.18. The isomorphism $\alpha : V' \times _{Y'} E \to W \times _ Z E$ determines an isomorphism $j_{small}^{-1}\mathcal{G}' \to g_{small}^{-1}\mathcal{G}$ of sheaves on $E$. By Lemma 81.4.7 we obtain a unique sheaf $\mathcal{F}$ on $Y$ pulling pack to $\mathcal{G}'$ and $\mathcal{G}$ compatibly with the isomorphism. By Properties of Spaces, Lemma 66.27.3 we obtain an étale morphism $V \to Y$ of algebraic spaces corresponding to $\mathcal{F}$; we omit the verification of the set theoretic condition¹. The given isomorphism $\mathcal{G}' \to f_{small}^{-1}\mathcal{F}$ and $\mathcal{G} \to i_{small}^{-1}\mathcal{F}$ corresponds to isomorphisms $V' \to V \times _ Y Y'$ and $W \to V \times _ Y Z$ compatible with $\alpha $ as desired. $\square$