
theorem Th5:
  for A,B being transitive with_units non empty AltCatStr, F1,F2
  being Functor of A,B holds F1 is_transformable_to F2 implies for t
  being transformation of F1,F2 holds (idt F2)`*`t = t & t`*`(idt F1) = t
proof
  let A,B be transitive with_units non empty AltCatStr, F1,F2 be
  Functor of A,B;
  assume
A1: F1 is_transformable_to F2;
  let t be transformation of F1,F2;
  now
    let a be Object of A;
A2: <^F1.a,F2.a^> <> {} by A1;
    thus ((idt F2)`*`t)!a = ((idt F2)!a)*(t!a) by A1,Def5
      .= (idm(F2.a))*(t!a) by Th4
      .= t!a by A2,ALTCAT_1:20;
  end;
  hence (idt F2)`*`t = t by A1,Th3;
  now
    let a be Object of A;
A3: <^F1.a,F2.a^> <> {} by A1;
    thus (t`*`(idt F1))!a = (t!a)*((idt F1)!a) by A1,Def5
      .= (t!a)*(idm(F1.a)) by Th4
      .= t!a by A3,ALTCAT_1:def 17;
  end;
  hence thesis by A1,Th3;
end;
