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