
theorem
  for C,D being category st C is initial & C ~= D holds D is initial
proof
  let C,D be category;
  assume
A1: C is initial;
  assume C ~= D;
  then consider F be Functor of C,D, G be Functor of D,C such that
A2: F is covariant & G is covariant & G (*) F = id C & F (*) G = id D
  by CAT_6:def 28;
  let B be category;
  consider F1 be Functor of C,B such that
A3: F1 is covariant &
  for G being Functor of C,B st G is covariant holds F1 = G by A1;
  set F2 = F1(*)G;
  take F2;
  for G1 being Functor of D,B st G1 is covariant holds F2 = G1
  proof
    let G1 be Functor of D,B;
    assume
A4: G1 is covariant;
    hence F2 = (G1(*)F)(*)G by A3,A2,CAT_6:35
    .= G1(*)(F(*)G) by A4,A2,CAT_7:10
    .= G1 by A2,A4,CAT_7:11;
  end;
  hence thesis by A2,A3,CAT_6:35;
end;
