Lemma 66.10.2. Let $S$ be a scheme. Let $j : X \to Y$ be a morphism of algebraic spaces over $S$. The following are equivalent:

$j$ is a monomorphism (as in Definition 66.10.1),

$j$ is a monomorphism in the category of algebraic spaces over $S$, and

the diagonal morphism $\Delta _{X/Y} : X \to X \times _ Y X$ is an isomorphism.

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