# Algebraic stacks: limit preserving versus locally of finite presentation

I'm wondering what the precise relationship is between an algebraic stack being locally of finite presentation and being limit preserving. Under some mild hypotheses on the diagonal (in force throughout the book), Prop. 4.18 of the book by Laumon--Moret-Bailly shows that a stack locally of finite presentation over some base scheme is limit preserving; I don't think the hypotheses on the diagonal are used here, and indeed I think I can piece together a proof pretty easily using the assumptions of the stacks project.

Now, I'm wondering if the converse is true. Some reasons to think it might be: in Artin's "Versal Deformations and Algebraic Stacks", he defines limit preserving, and says that it is what he had previously referred to as locally of finite presentation. Also, for algebraic spaces we have http://stacks.math.columbia.edu/tag/05N0 which shows that the two conditions are equivalent (and equivalent to being limit preserving on objects, unsurprisingly).

A reason to think they might not be equivalent: I haven't seen this statement anywhere in the literature.

Suppose that $\mathcal{X}$ is an algebraic stack which is limit preserving on objects over $S$. Suppose that $U \to \mathcal{X}$ is a smooth surjective map from a scheme. Then $U \to S$ is limit preserving by the results of Section Tag 06CT. Thus $U$ is locally of finite presentation over S (for example by the reference you gave). This exactly means that $\mathcal{X} \to S$ is locally of finite presentation.