  We formulate some properties of a conjectural object $X_{fun}(r,n)$
parametrizing Anderson t-motives of dimension $n$ and rank $r$. Namely, we give
formulas for $\goth p$-Hecke correspondences of $X_{fun}(r,n)$ and its
reductions at $\goth p$ (where $\goth p$ is a prime of $\Bbb F_q[\theta]$).
Also, we describe their geometric interpretation. These results are analogs of
the corresponding results of reductions of Shimura varieties. Finally, we give
conjectural formulas for Hodge numbers (over the fields generated by Hecke
correspondences) of middle cohomology submotives of $X_{fun}(r,n)$.
