reserve x,y for set;

theorem
  for A,B being category, F being covariant Functor of A,B st F is
  bijective holds A, B are_isomorphic_under F
proof
  let A,B be category, F be covariant Functor of A,B such that
A1: F is bijective;
  the Arrows of A = the Arrows of A & the Arrows of B = the Arrows of B;
  hence A is subcategory of A & B is subcategory of B by ALTCAT_2:20
,ALTCAT_4:35;
  take F;
  thus F is bijective & for a9 being Object of A, a being Object of A st a9 =
  a holds F.a9 = F.a by A1;
  let b9,c9 be Object of A, b,c be Object of A;
  assume
A2: <^b9,c9^> <> {} & b9 = b & c9 = c;
  then <^F.b,F.c^> <> {} by FUNCTOR0:def 18;
  hence thesis by A2,FUNCTOR0:def 15;
end;
