reserve x,y for set;

theorem
  for A,B being category, F being covariant Functor of A,B st A, B
  are_isomorphic_under F holds F is bijective
proof
  let A,B be category, F be covariant Functor of A,B such that
  A is subcategory of A and
  B is subcategory of B;
  given G being covariant Functor of A,B such that
A1: G is bijective and
A2: for a9 being Object of A, a being Object of A st a9 = a holds G.a9 =
  F.a and
A3: for b9,c9 being Object of A, b,c being Object of A st <^b9,c9^> <>
{} & b9 = b & c9 = c for f9 being Morphism of b9,c9, f being Morphism of b,c st
  f9 = f holds G.f9 = Morph-Map(F,b,c).f;
  G is injective surjective by A1;
  then
A4: G is one-to-one faithful full onto;
A5: now
    let a,b be Object of A such that
A6: <^a,b^> <> {};
    let f be Morphism of a,b;
    <^F.a, F.b^> <> {} by A6,FUNCTOR0:def 18;
    hence F.f = Morph-Map(F,a,b).f by A6,FUNCTOR0:def 15
      .= G.f by A3,A6;
  end;
  for a being Object of A holds F.a = G.a by A2;
  then the FunctorStr of F = the FunctorStr of G by A5,YELLOW18:1;
  hence the ObjectMap of F is one-to-one & the MorphMap of F is "1-1" & (ex f
being ManySortedFunction of (the Arrows of A), (the Arrows of B)*the ObjectMap
  of F st f = the MorphMap of F & f is "onto") & the ObjectMap of F is onto by
A4;
end;
