Suppose I have an Artin stack $\mathfrak M$ locally of finite-type over $\mathbb C$ with presentation $M\rightarrow \mathfrak M$. Suppose further that $\mathfrak M$ "represents" (in the stack sense) some set of objects which are bounded, i.e. roughly there is a family of such objects $\mathcal F$ over a scheme $S$ of finite type over $\mathbb C$ such that every element of our set of objects occurs as $\mathcal F_s$ for some closed points $s\in S$. Why does it follow that $\mathfrak M$ is actually of finite-type over $\mathbb C$?
Specifically, I've seen it argued that $S$ induces a surjection $S\rightarrow M$ of schemes over $\mathbb C$, which thus makes $M$ of finite-type over $\mathbb C$. But I don't see why we get an induced map to $M$ instead of just to $\mathfrak M$. Indeed the family $\mathcal F$ should induce a surjective morphism $S\rightarrow \mathfrak M$, but I don't see why this should factor through $M$.