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 Th48:
  for F being Functor of A,B, G being Functor of B,A st G*F ~= id
  A holds F is faithful
proof
  let F be Functor of A,B, G be Functor of B,A;
  assume G*F ~= id A;
  then
A1: id A ~= G*F by NATTRA_1:28;
  then
A2: id A is_naturally_transformable_to G*F;
  consider s being natural_transformation of id A, G*F such that
A3: s is invertible by A1;
  thus F is faithful
  proof
    let a,a9 be Object of A;
    assume
A4: Hom(a,a9) <> {};
    then
A5: Hom((id A).a,(id A).a9) <> {} by CAT_1:84;
    let f1,f2 be Morphism of a,a9;
    assume
A6: F.(f1 qua Morphism of A) = F.(f2 qua Morphism of A);
A7: (G*F)/.f1 = (G*F).(f1 qua Morphism of A) by A4,CAT_3:def 10
      .= G.(F.(f2 qua Morphism of A)) by A6,FUNCT_2:15
      .= (G*F).(f2 qua Morphism of A) by FUNCT_2:15
      .= (G*F)/.f2 by A4,CAT_3:def 10;
A8: s.a9*(id A)/.f1 = (G*F)/.f1*s.a by A2,A4,NATTRA_1:def 8
      .= s.a9*(id A)/.f2 by A2,A4,A7,NATTRA_1:def 8;
    s.a9 is invertible by A3;
    then
A9: s.a9 is monic by CAT_1:43;
    thus f1 = (id A)/.f1 by A4,NATTRA_1:16
      .= (id A)/.f2 by A5,A9,A8
      .= f2 by A4,NATTRA_1:16;
  end;
end;
