Lemma 87.27.4. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of formal algebraic spaces over $S$. The following are equivalent

$f$ is a closed immersion,

for every scheme $T$ and morphism $T \to Y$ the projection $X \times _ Y T \to T$ is a closed immersion,

for every affine scheme $T$ and morphism $T \to Y$ the projection $X \times _ Y T \to T$ is a closed immersion,

there exists a covering $\{ Y_ j \to Y\} $ as in Definition 87.11.1 such that each $X \times _ Y Y_ j \to Y_ j$ is a closed immersion, and

there exists a morphism $Z \to Y$ of formal algebraic spaces which is representable by algebraic spaces, surjective, flat, and locally of finite presentation such that $X \times _ Y Z \to X$ is a closed immersion, and

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