reserve y for set;
reserve A for Category,
  a,o for Object of A;
reserve f for Morphism of A;

theorem
  for F being Functor of A,Functors(A,EnsHom(A)) st Obj F is one-to-one
  & F is faithful holds F is one-to-one
proof
  let F be Functor of A,Functors(A,EnsHom(A));
  assume
A1: Obj F is one-to-one;
  assume
A2: F is faithful;
  for x1,x2 be object st x1 in dom F & x2 in dom F & F.x1 = F.x2 holds x1 = x2
  proof
    let x1,x2 be object;
    assume that
A3: x1 in dom F & x2 in dom F and
A4: F.x1 = F.x2;
    reconsider m1=x1,m2=x2 as Morphism of A by A3,FUNCT_2:def 1;
    set o1=dom m1,o2=cod m1;
    set o3=dom m2,o4=cod m2;
    reconsider m19=m1 as Morphism of o1,o2 by CAT_1:4;
    reconsider m29=m2 as Morphism of o3,o4 by CAT_1:4;
A5: Hom(o1,o2) <> {} by CAT_1:2;
    then
A6: Hom(F.o1,F.o2) <> {} by CAT_1:84;
A7: Hom(o3,o4) <> {} by CAT_1:2;
    then
A8: Hom(F.o3,F.o4) <> {} by CAT_1:84;
A9: F/.m19= F.m2 by A4,A5,CAT_3:def 10
      .= F/.m29 by A7,CAT_3:def 10;
    (Obj F).o1 = F.o1
      .=dom (F/.m29) by A9,A6,CAT_1:5
      .= (Obj F).o3 by A8,CAT_1:5;
    then
A10: m2 is Morphism of dom m2,cod m2 & o1=o3 by A1,CAT_1:4,FUNCT_2:19;
    (Obj F).o2 = F.o2
      .=cod(F/.m29) by A9,A6,CAT_1:5
      .=(Obj F).o4 by A8,CAT_1:5;
    then m1 is Morphism of dom m1,cod m1 & m2 is Morphism of o1,o2 by A1,A10,
CAT_1:4,FUNCT_2:19;
    hence thesis by A2,A4,A5;
  end;
  hence thesis by FUNCT_1:def 4;
end;
