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 = idt F1
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" `*` (e qua transformation of F1, F2))!a by A3,
FUNCTOR2:def 8
      .= (e"!a)*(e!a) by A2,FUNCTOR2:def 5
      .= (e!a)"*(e!a) by A1,Th38
      .= idm (F1.a) by A4,ALTCAT_3:def 5
      .= (idt F1)!a by FUNCTOR2:4;
  end;
  hence thesis by FUNCTOR2:3;
end;
