
theorem
  for C1,C2,D being category, P1 being Functor of D,C1,
      P2 being Functor of D,C2
  st P1 is covariant & P2 is covariant &
  D,P1,P2 is_product_of C1,C2 holds D,P2,P1 is_product_of C2,C1
  proof
    let C1,C2,D be category;
    let P1 be Functor of D,C1;
    let P2 be Functor of D,C2;
    assume
A1: P1 is covariant & P2 is covariant;
    assume
A2: D,P1,P2 is_product_of C1,C2;
    for D1 being category, G1 being Functor of D1,C2, G2 being Functor of D1,C1
    st G1 is covariant & G2 is covariant holds
    ex H being Functor of D1,D st H is covariant &
    P2 (*) H = G1 & P1 (*) H = G2 & for H1 being Functor of D1,D
    st H1 is covariant & P2 (*) H1 = G1 & P1 (*) H1 = G2 holds H = H1
    proof
      let D1 be category;
      let G1 be Functor of D1,C2;
      let G2 be Functor of D1,C1;
      assume
A3:   G1 is covariant & G2 is covariant;
      consider H be Functor of D1,D such that
A4:   H is covariant & P1 (*) H = G2 & P2 (*) H = G1
      & for H1 being Functor of D1,D
      st H1 is covariant & P1 (*) H1 = G2 & P2 (*) H1 = G1 holds H = H1
      by A3,A2,A1,Def17;
      take H;
      thus H is covariant & P2 (*) H = G1 & P1 (*) H = G2 by A4;
      let H1 be Functor of D1,D;
      assume H1 is covariant & P2 (*) H1 = G1 & P1 (*) H1 = G2;
      hence H = H1 by A4;
    end;
    hence D,P2,P1 is_product_of C2,C1 by A1,Def17;
  end;
