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Morphism from projective space to product of projective spaces

I have a problem with creating a rational map from the projective plane to P^1xP^1. The following code gives the error "polys (=[x, y, x^2, y^2]) must be of the same degree":

K = GF(2)
P2.<x,y,z> = ProjectiveSpace(K,2)
P1P1.<x0,x1,y0,y1> = ProductProjectiveSpaces([1,1],K)
H = Hom(P2,P1P1)

On the other hand, creating the "same" map on P^1xP^1 does not give an error:

E = End(P1P1)

Can someone explain to me why it does not work and how I can go around this problem?