
theorem Th26:
  for C,D being category st C is terminal & C ~= D holds D is terminal
proof
  let C,D be category;
  assume
A1: C is terminal;
  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 B,C such that
A3: F1 is covariant &
  for G being Functor of B,C st G is covariant holds F1 = G by A1;
  set F2 = F(*)F1;
  take F2;
  for G1 being Functor of B,D st G1 is covariant holds F2 = G1
  proof
    let G1 be Functor of B,D;
    assume
A4: G1 is covariant;
    hence F2 = F(*)(G(*)G1) by A3,A2,CAT_6:35
    .= (F(*)G)(*)G1 by A4,A2,CAT_7:10
    .= G1 by A2,A4,CAT_7:11;
  end;
  hence thesis by A2,A3,CAT_6:35;
end;
