reserve A for non empty AltCatStr,
  B, C for non empty reflexive AltCatStr,
  F for feasible Covariant FunctorStr over A, B,
  G for feasible Covariant FunctorStr over B, C,
  M for feasible Contravariant FunctorStr over A, B,
  N for feasible Contravariant FunctorStr over B, C,
  o1, o2 for Object of A,
  m for Morphism of o1, o2;
reserve A, B, C, D for transitive with_units non empty AltCatStr,
  F1, F2, F3 for covariant Functor of A, B,
  G1, G2, G3 for covariant Functor of B, C,
  H1, H2 for covariant Functor of C, D,
  p for transformation of F1, F2,
  p1 for transformation of F2, F3,
  q for transformation of G1, G2,
  q1 for transformation of G2, G3,
  r for transformation of H1, H2;
reserve A, B, C, D for category,
  F1, F2, F3 for covariant Functor of A, B,
  G1, G2, G3 for covariant Functor of B, C;
reserve t for natural_transformation of F1, F2,
  s for natural_transformation of G1, G2,
  s1 for natural_transformation of G2, G3;
reserve e for natural_equivalence of F1, F2,
  e1 for natural_equivalence of F2, F3,
  f for natural_equivalence of G1, G2;

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;
