reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;
reserve F,F1,F2,F3 for Functor of A,B,
  G,G1,G2,G3 for Functor of B,C,
  H,H1,H2 for Functor of C,D,
  s for natural_transformation of F1,F2,
  s9 for natural_transformation of F2,F3,
  t for natural_transformation of G1,G2,
  t9 for natural_transformation of G2,G3,
  u for natural_transformation of H1,H2;

theorem
  A,B are_equivalent implies for F being Equivalence of A,B holds F is
full & F is faithful & for b being Object of B ex a being Object of A st b, F.a
  are_isomorphic
proof
  assume
A1: A,B are_equivalent;
  let F be Equivalence of A,B;
  consider G being Equivalence of B,A such that
A2: G*F ~= id A and
A3: F*G ~= id B by A1,Th47;
A4: id A ~= G*F by A2,NATTRA_1:28;
  then
A5: id A is_naturally_transformable_to G*F;
  consider s being natural_transformation of id A, G*F such that
A6: s is invertible by A4;
A7: G is faithful by A3,Th48;
  thus F is full
  proof
    let a,a9 be Object of A such that
A8: Hom(F.a,F.a9) <> {};
    reconsider f2 = s.a9 as Morphism of a9, (G*F).a9 by CAT_1:79;
    reconsider f1 = s.a as Morphism of a, (G*F).a by CAT_1:79;
A9: s.a9 is invertible by A6;
A10: Hom((id A).a9,(G*F).a9) <> {} by A5,Th23;
    let g be Morphism of F.a,F.a9;
A11: (G*F).a = G.(F.a) by CAT_1:76;
    then reconsider h = G/.g as Morphism of (G*F).a, (G*F).a9 by CAT_1:76;
A12: Hom((id A).a,(G*F).a) <> {} by A5,Th23;
    (G*F).a9 = G.(F.a9) by CAT_1:76;
    then
A13: Hom((G*F).a,(G*F).a9) <> {} by A8,A11,CAT_1:84;
    then
A14: Hom((id A).a,(G*F).a9) <> {} by A12,CAT_1:24;
    s.a is invertible by A6;
    then
A15: s.a is epi by CAT_1:43;
    (G*F) is_naturally_transformable_to id A by A2;
    then
A16: Hom((G*F).a9,(id A).a9) <> {} by Th23;
A17: (id A).a = a & (id A).a9 = a9 by CAT_1:79;
    hence
A18: Hom(a,a9) <> {} by A16,A14,CAT_1:24;
    take f = f2"*(h*f1);
A19: (id A)/.f = ((s.a9)"*(h*(s.a))) by A17,A18,NATTRA_1:16;
    h*s.a = id((G*F).a9)*(h*(s.a)) by A14,CAT_1:28
      .= s.a9*(s.a9)"*(h*(s.a)) by A9,CAT_1:def 17
      .= s.a9*(id A)/.f by A16,A14,A10,A19,CAT_1:25
      .= (G*F)/.f*s.a by A5,A18,NATTRA_1:def 8;
    then
A20: h = (G*F)/.f by A13,A15;
    G.(g qua Morphism of B) = G/.g by A8,CAT_3:def 10
      .= (G*F).(f qua Morphism of A) by A18,A20,CAT_3:def 10
      .= G.(F.(f qua Morphism of A)) by FUNCT_2:15
      .= G.(F/.f qua Morphism of B) by A18,CAT_3:def 10;
    hence g = F/.f by A7,A8
      .= F.(f qua Morphism of A) by A18,CAT_3:def 10;
  end;
  thus F is faithful by A2,Th48;
  let b be Object of B;
  take G.b;
A21: F.(G.b) = (F*G).b & (id B).b = b by CAT_1:76,79;
A22: id B ~= F*G by A3,NATTRA_1:28;
  then id B is_naturally_transformable_to F*G;
  then
A23: id B is_transformable_to F*G;
  ex t being natural_transformation of id B, F*G st t is invertible by A22;
  hence thesis by A21,A23,Th4;
end;
