Let $S$ be a Noetherian scheme and $X$ a projective $S$-scheme.

Reading Laumon-Moret--Bailly's "Champs Algebriques", Theorem 4.6.2.1:

$ \mathscr Coh_{X|S} $ and $ \mathscr Fib_{X|S,r} $ are algebraic stacks of finite type.

The proof of "finite type" part seems to be done by proving that the $\mathfrak Isom$ sheaf above $U\in \operatorname{Aff}/S$ is a separated $U$-scheme of finite type.

Is there any reference that this property of the $\mathfrak Isom$ sheaf implies that the stack is of finite type?

I know that geometric properties of the algebraic stacks are "transferred" to stacks by a presentation/atlas, but I am not aware of such a thing for the $\mathfrak Isom$ sheaf.

Thank you for any reference on the subject.