Lemma 105.4.1 (Key fact). The functor $\mathit{Sch}^{opp} \to \textit{Sets}$, $T \mapsto \{ (a, a', \alpha )\text{ as above}\} $ is representable by a scheme $S \times _{\mathcal{M}_{1, 1}} S'$.

**Proof.**
Idea of proof. Relate this functor to

\[ \mathit{Isom}_{S \times S'}(E \times S', S \times E') \]

and use Grothendieck's theory of Hilbert schemes. $\square$

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