theorem
  F1, F2 are_naturally_equivalent implies (e")" = e
proof
  assume
A1: F1, F2 are_naturally_equivalent;
  then
A2: F1 is_transformable_to F2 by Def4;
  now
    let a be Object of A;
A3: <^F1.a,F2.a^> <> {} by A2;
    F2 is_transformable_to F1 by A1;
    then
A4: <^F2.a,F1.a^> <> {};
    e!a is iso by A1,Def5;
    then
A5: e!a is retraction coretraction by ALTCAT_3:5;
    thus (e")"!a = (e"!a)" by A1,Th38
      .= ((e!a)")" by A1,Th38
      .= e!a by A3,A4,A5,ALTCAT_3:3;
  end;
  hence thesis by A2,FUNCTOR2:3;
end;
