reserve x,y for set;

theorem
  for A,B being category, F being contravariant Functor of A,B st F is
bijective for G being contravariant Functor of B,A st (for b being Object of B
  holds F.(G.b) = b) & (for a,b being Object of B st <^a,b^> <> {} for f being
  Morphism of a,b holds F.(G.f) = f) holds the FunctorStr of G = F"
proof
  let A,B be category, F be contravariant Functor of A,B such that
A1: F is bijective;
  let G be contravariant Functor of B,A such that
A2: for b being Object of B holds F.(G.b) = b and
A3: for a,b being Object of B st <^a,b^> <> {} for f being Morphism of a
  ,b holds F.(G.f) = f;
A4: now
    let a,b be Object of B such that
A5: <^a,b^> <> {};
    let f be Morphism of a,b;
    thus (F*G).f = F.(G.f) by A5,FUNCTOR3:7
      .= f by A3,A5
      .= (id B).f by A5,FUNCTOR0:31;
  end;
  now
    let b be Object of B;
    thus (F*G).b = F.(G.b) by FUNCTOR0:33
      .= b by A2
      .= (id B).b by FUNCTOR0:29;
  end;
  hence thesis by A1,A4,Th4,YELLOW18:1;
end;
