reserve A,B,C,D for Category,
  F for Functor of A,B,
  G for Functor of B,C;
reserve o,m for set;

theorem Th7:
  F is one-to-one implies Obj F is one-to-one
proof
  assume
A1: F is one-to-one;
  let x1,x2 be object;
  assume x1 in dom Obj F & x2 in dom Obj F;
  then reconsider a1 = x1, a2 = x2 as Object of A;
  assume (Obj F).x1 = (Obj F).x2;
  then F.a1 = F.a2;
  then
A2: F.(id a1 qua Morphism of A) = id(F.a2) by CAT_1:71
    .= F.(id a2 qua Morphism of A)by CAT_1:71;
  dom F = the carrier' of A by FUNCT_2:def 1;
  then id a1 = id a2 by A1,A2;
  hence thesis by CAT_1:59;
end;
