
theorem
  for C,D being Category holds [(the carrier of C) --> D, (the carrier'
of C) --> id D] is Indexing of C & [(the carrier of C) --> D, (the carrier' of
  C) --> id D] is coIndexing of C
proof
  let C,D be Category;
  set H = [(the carrier of C) --> D, (the carrier' of C) --> id D], I = H;
A1: for a be Object of C holds H`2.id a = id (H`1.a);
  for m being Morphism of C holds H`2.m is Functor of (H`1*(the Source of
  C)).m, (H`1*(the Target of C)).m by Lm4;
  then
  H`2 is ManySortedFunctor of H`1*(the Source of C), H`1*(the Target of C)
  by Def7;
  then reconsider H as Indexing of the Source of C, the Target of C by Def8;
  
  for m1, m2 be Morphism of C st dom m2 = cod m1 holds
  H`2.(m2(*)m1) = (H`2.m2)*(H`2.m1) by FUNCT_2:17;
  hence I is Indexing of C by A1,Th6;
  for m being Morphism of C holds H`2.m is Functor of (H`1*(the Target of
  C)).m, (H`1*(the Source of C)).m by Lm5;
  then
  H`2 is ManySortedFunctor of H`1*(the Target of C), H`1*(the Source of C
  ) by Def7;
  then reconsider H as Indexing of the Target of C, the Source of C by Def8;
  for m1, m2 be Morphism of C st dom m2 = cod m1 holds
   H`2.(m2(*)m1) = (H`2.m1)*(H`2.m2) by FUNCT_2:17;
  hence thesis by A1,Th7;
end;
