
theorem Th15:
  for A, B being category st A, B are_opposite
  holds dualizing-func(A,B) is bijective
proof
  let A, B be category such that
A1: A, B are_opposite;
  set F = dualizing-func(A,B);
  deffunc O(set) = $1;
  deffunc F(set,set,set) = $3;
A2: for a being Object of A holds F.a = O(a) by A1,Def5;
A3: for a,b being Object of A st <^a,b^> <> {}
  for f being Morphism of a,b holds F.f = F(a,b,f) by A1,Def5;
A4: for a,b being Object of A st O(a) = O(b) holds a = b;
A5: for a,b being Object of A st <^a,b^> <> {}
  for f,g being Morphism of a,b st F(a,b,f) = F(a,b,g) holds f = g;
A6: now
    let a,b be Object of B;
    reconsider a9 = a, b9 = b as Object of A by A1;
A7: <^a,b^> = <^b9,a9^> by A1,Th9;
    assume
A8: <^a,b^> <> {};
    let f be Morphism of a,b;
    thus ex c,d being Object of A, g being Morphism of c,d
    st b = O(c) & a = O(d) & <^c,d^> <> {} & f = F(c,d,g) by A7,A8;
  end;
  F is bijective from ContraBijectiveSch(A2,A3,A4,A5,A6);
  hence thesis;
end;
