reserve C for category,
  o1, o2, o3 for Object of C;

theorem
  for A, B being category, F being contravariant Functor of A, B for o1,
o2 being Object of A st F is full faithful & <^o1,o2^> <> {} & <^o2,o1^> <> {}
  & F.o2, F.o1 are_iso holds o1, o2 are_iso
proof
  let A, B be category, F be contravariant Functor of A, B, o1, o2 be Object
  of A such that
A1: F is full faithful and
A2: <^o1,o2^> <> {} and
A3: <^o2,o1^> <> {} and
A4: F.o2, F.o1 are_iso;
  consider Fa being Morphism of F.o2, F.o1 such that
A5: Fa is iso by A4;
  consider a being Morphism of o1, o2 such that
A6: Fa = F.a by A1,A2,Th17;
  thus <^o1,o2^> <> {} & <^o2,o1^> <> {} by A2,A3;
  take a;
  thus thesis by A1,A2,A3,A5,A6,Th32;
end;
