
theorem Th14:
  for A,B being category st A,B are_opposite
  holds dualizing-func(A,B)*dualizing-func(B,A) = id B
proof
  let A,B be category such that
A1: A, B are_opposite;
A2: now
    let a be Object of B;
    thus (dualizing-func(A,B)*dualizing-func(B,A)).a
    = dualizing-func(A,B).(dualizing-func(B,A).a) by FUNCTOR0:33
      .= dualizing-func(B,A).a by A1,Def5
      .= a by A1,Def5
      .= (id B).a by FUNCTOR0:29;
  end;
  now
    let a,b be Object of B;
    assume
A3: <^a,b^> <> {};
    then
A4: <^dualizing-func(B,A).b,dualizing-func(B,A).a^> <> {} by FUNCTOR0:def 19;
    let f be Morphism of a,b;
    thus (dualizing-func(A,B)*dualizing-func(B,A)).f
    = dualizing-func(A,B).(dualizing-func(B,A).f) by A3,FUNCTOR3:7
      .= dualizing-func(B,A).f by A1,A4,Def5
      .= f by A1,A3,Def5
      .= (id B).f by A3,FUNCTOR0:31;
  end;
  hence thesis by A2,Th1;
end;
