RationalMap * RationalMap -- composition of rational maps

i1 : R = QQ[x_0..x_3]; S = QQ[y_0..y_4]; T = QQ[z_0..z_4];

i4 : phi = rationalMap(R,S,{x_0*x_2,x_0*x_3,x_1*x_2,x_1*x_3,x_2*x_3})

o4 = -- rational map --
     source: Proj(QQ[x , x , x , x ])
                      0   1   2   3
     target: Proj(QQ[y , y , y , y , y ])
                      0   1   2   3   4
     defining forms: {
                      x x ,
                       0 2
                      
                      x x ,
                       0 3
                      
                      x x ,
                       1 2
                      
                      x x ,
                       1 3
                      
                      x x
                       2 3
                     }

o4 : RationalMap (quadratic rational map from PP^3 to PP^4)

i5 : psi = rationalMap(S,T,{y_0*y_3,-y_2*y_3,y_1*y_2,y_2*y_4,-y_3*y_4})

o5 = -- rational map --
     source: Proj(QQ[y , y , y , y , y ])
                      0   1   2   3   4
     target: Proj(QQ[z , z , z , z , z ])
                      0   1   2   3   4
     defining forms: {
                      y y ,
                       0 3
                      
                      -y y ,
                        2 3
                      
                      y y ,
                       1 2
                      
                      y y ,
                       2 4
                      
                      -y y
                        3 4
                     }

o5 : RationalMap (quadratic rational map from PP^4 to PP^4)

i6 : phi * psi

o6 = -- rational map --
     source: Proj(QQ[x , x , x , x ])
                      0   1   2   3
     target: Proj(QQ[z , z , z , z , z ])
                      0   1   2   3   4
     defining forms: {
                      x ,
                       0
                      
                      -x ,
                        1
                      
                      x ,
                       0
                      
                      x ,
                       2
                      
                      -x
                        3
                     }

o6 : RationalMap (linear rational map from PP^3 to PP^4)

i7 : (map phi) * (map psi)

                             2                   2          2
o7 = map (R, T, {x x x x , -x x x , x x x x , x x x , -x x x })
                  0 1 2 3    1 2 3   0 1 2 3   1 2 3    1 2 3

o7 : RingMap R <-- T