  In this paper we first review the setting for the geometric Langlands
functoriality and establish a result for the `backward' functoriality functor.
We illustrate this by known examples of the geometric theta-lifting. We then
apply the above result to obtain new Hecke eigen-sheaves. The most important
application is a construction of the automorphic sheaf for G=GSp_4 attached to
a G^L-local system on a curve X such that its standard representation is an
irreducible local system of rank 4 on X.
