reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;

theorem Th1:
  for a being Object of A holds [[id a,id a],id a] in the Comp of A
proof
  let a be Object of A;
A1: dom id a = a;
A2: cod id a = a;
  then
A3: [id a,id a] in dom the Comp of A by A1,CAT_1:15;
  (id a)(*)(id a) = id a by A1,CAT_1:22;
  then (the Comp of A).(id a,id a) = id a by A1,A2,CAT_1:16;
  hence thesis by A3,FUNCT_1:def 2;
end;
