Assume that I have two moduli spaces $M_1,M_2$ solving two problems $F_1,F_2:C\to Sets$, think for example about $C=Sch/S$. Furthermore I have a morphism $f:M_1\to M_2$ or equivalently a natural transformation $\alpha_f:F_1\to F_2$.
I would like to know how to relate geometric properties of $f$ (e.g. being etale,smooth,open,closed...) with the abstract properties of $\alpha_f$.
For example take the case where $\alpha_f$ "forgets" some part of the moduli problem; even more concretely assume $F_1(X)=\{A/X$ an abelian variety with some extra structure 1 and extra structure 2$\}$ and $F_2(X)=\{A/X$ an abelian variety with extra structure 1$\}$, then I have the natural transformation $\alpha_f:F_1\to F_2$ corresponding to forgetting the second extra structure. In this example what can I conclude about the corresponding morphism $f:M_1\to M_2$ (and why)? My intuition tells me that it should be open, but I can't find a rigorous argument for that.