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
  for c being Object of A, d being Object of B st F/.(id c) = id d
   holds F.c = d
proof
  let c be Object of A, d be Object of B;
A1: Hom(c,c) <> {};
  assume F/.(id c) = id d;
  then F.(id c qua Morphism of A) = id d by A1,CAT_3:def 10;
  hence thesis by CAT_1:70;
end;
