
theorem
  for C being composable with_identities non empty CategoryStr,
      a,b being Object of C
  holds a is initial & b is initial implies a,b are_isomorphic
proof
  let C be composable with_identities non empty CategoryStr,
      a,b be Object of C;
  assume that
A1: a is initial and
A2: b is initial;
  set g = the Morphism of b,a;
  set f = the Morphism of a,b;
A3: Hom(b,a) <> {} by A2;
  f is isomorphism
  proof
A4: Hom(a,b) <> {} by A1;
    g*f = id- a & f*g = id- b by A1,A2,Th31;
    hence thesis by A3,A4,CAT_7:def 9;
  end;
  hence thesis by CAT_7:def 10;
end;
