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.