theorem
  a is initial iff for b ex f being Morphism of a,b st Hom(a,b) = {f}
proof
  thus a is initial implies for b ex f being Morphism of a,b st Hom(a,b) = {f}
  proof
    assume
A1: a is initial;
    let b;
    consider f being Morphism of a,b such that
A2: for g being Morphism of a,b holds f = g by A1;
    take f;
    thus thesis by A2,Th7,A1;
  end;
  assume
A3: for b ex f being Morphism of a,b st Hom(a,b) = {f};
  let b;
  consider f being Morphism of a,b such that
A4: Hom(a,b) = {f} by A3;
  thus Hom(a,b) <> {} by A4;
  take f;
  thus thesis by A4,Th6;
end;
