Lemma 87.21.5. Let $S$ be a scheme. Let $f : X \to Y$ and $g : Y \to Z$ be morphisms locally of finite type of locally Noetherian formal algebraic spaces over $S$. If $g \circ f$ is rig-surjective and $g$ is a monomorphism, then $f$ is rig-surjective.

**Proof.**
Use Lemma 87.21.4 and that $f$ is the base change of $g \circ f$ by $g$.
$\square$

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