
theorem
  for C being Category, D being Categorial Category, F being Functor of
  C,D holds D is TargetCat of F-indexing_of C
proof
  let C be Category, D be Categorial Category, F be Functor of C,D;
  set I = F-indexing_of C;
  thus for a being Object of C holds I`1.a is Object of D by FUNCT_2:5;
  let b be Morphism of C;
  set h = F.b;
A1: dom h = h`11 by CAT_5:13;
  cod h = h`12 by CAT_5:13;
  then consider f being Functor of h`11, h`12 such that
A2: h = [[h`11, h`12], f] by A1,CAT_5:def 6;
A3: cod h = (Obj F).cod b by CAT_1:69
    .= (Obj F).((the Target of C).b);
A4: dom h = (Obj F).dom b by CAT_1:69
    .= (Obj F).((the Source of C).b);
  I`2 = pr2 F & dom F = the carrier' of C by FUNCT_2:def 1;
  then I`2.b = h`2 by MCART_1:def 13
    .= f by A2;
  hence thesis by A1,A2,A4,A3,CAT_5:13;
end;
