ag.algebraic geometry - Is a sub-stack of a scheme a scheme?

Let $f:mathcal{X}rightarrow Y$ be a morphism from an Artin stack to a scheme such that $f$ is an immersion. Then $mathcal{X}$ is automatically an algebraic space, so we're done by Knutson, Algebraic spaces, II.6.16.

Additions prompted by Brian's comment

Assume that $f:mathcal{X}rightarrow Y$ is a schematic map, and that $Y$ is a scheme; then $f$ is the pullback of $f$ over the map of schemes $mathrm{id}_Y$, so $mathcal{X}$ must be a scheme. Knutson needs lemma II.6.16 because he doesn't use the now-standard definition of schematic, but atlases instead.

When using immersion, I always mean $jcirc i$, where $i$ is a closed immersion and $j$ an open one, following EGA I. But I understand that this is not a better choice than the other way round, and that they are only equivalent when the morphism is quasicompact.

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