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Try to upgrade your version of Sage. This works fine in Sage 9.1:

sage: K = GF(2)
....: P2.<x,y,z> = ProjectiveSpace(K,2)
....: P1P1.<x0,x1,y0,y1> = ProductProjectiveSpaces([1,1],K)
....: H = Hom(P2,P1P1)
....: H([x,y,x^2,y^2])
....:
Scheme morphism:
  From: Projective Space of dimension 2 over Finite Field of size 2
  To:   Product of projective spaces P^1 x P^1 over Finite Field of size 2
  Defn: Defined on coordinates by sending (x : y : z) to
        (x : y , x^2 : y^2)