Lemma 87.15.5. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The diagonal morphism $\Delta : X \to X \times _ Y X$ is representable (by schemes), a monomorphism, locally quasi-finite, locally of finite type, and separated.

**Proof.**
Let $T$ be a scheme and let $T \to X \times _ Y X$ be a morphism. Then

\[ T \times _{(X \times _ Y X)} X = T \times _{(X \times _ S X)} X \]

Hence the result follows immediately from Lemma 87.11.2. $\square$

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