theorem Th39:
  F1, F2 are_naturally_equivalent implies e `*` e" = idt F2
proof
  assume
A1: F1, F2 are_naturally_equivalent;
  then
A2: F1 is_transformable_to F2 & F2 is_transformable_to F1 by Def4;
A3: F1 is_naturally_transformable_to F2 & F2 is_naturally_transformable_to
  F1 by A1,Def4;
  now
    let a be Object of A;
A4: e!a is iso by A1,Def5;
    thus (e `*` e")!a = ((e qua transformation of F1, F2) `*` e")!a by A3,
FUNCTOR2:def 8
      .= (e!a)*(e"!a) by A2,FUNCTOR2:def 5
      .= (e!a)*((e!a)") by A1,Th38
      .= idm (F2.a) by A4,ALTCAT_3:def 5
      .= (idt F2)!a by FUNCTOR2:4;
  end;
  hence thesis by FUNCTOR2:3;
end;
