## moduli problems and morphisms

Hello,

I'm studying the reduction of two Shimura varieties over $\overline{\mathbb{F}_p}$. I define a map on the moduli problems of these two varieties, and I would like to show that this is actually an algebraic map between algebraic varieties.

I need to define a morphism of functors between the two moduli problems, but this seems to require an extensive use of Dieudonné module theory over any $\overline{\mathbb{F}_p}$-scheme, which is too complicated in this case. Is there any other option ?

For example, in the proof of the Eichler-Shimura relation for modular curves, one defines sections for the two maps $X_0(p) \longrightarrow X_0$. Why are they algebraic maps ? Thank you !

-
 Just to clarify - are you asking how to show that your set-theoretic map is algebraic without showing that it arises from a morphism of functors? Or why morphisms of functors really do give algebraic maps? – Ramsey Nov 7 2011 at 17:25