reserve x,y for set;

theorem
  for A1,A2 being category, F being contravariant Functor of A1,A2 st F
  is bijective for B1 being non empty subcategory of A1 for B2 being non empty
  subcategory of A2 st B1,B2 are_anti-isomorphic_under F holds B2,B1
  are_anti-isomorphic_under F"
proof
  let A1,A2 be category, F be contravariant Functor of A1,A2 such that
A1: F is bijective;
  F is surjective by A1;
  then F is onto;
  then
A2: F is coreflexive by FUNCTOR0:47;
  ex H being Functor of A2,A1 st H = F" & H is bijective contravariant by A1,
FUNCTOR0:49;
  then reconsider F9 = F" as contravariant Functor of A2,A1;
  let B1 be non empty subcategory of A1;
  let B2 be non empty subcategory of A2 such that
  B1 is subcategory of A1 and
  B2 is subcategory of A2;
  given G being contravariant Functor of B1,B2 such that
A3: G is bijective and
A4: for a being Object of B1, a1 being Object of A1 st a = a1 holds G.a
  = F.a1 and
A5: for b,c being Object of B1, b1,c1 being Object of A1 st <^b,c^> <>
{} & b = b1 & c = c1 for f being Morphism of b,c, f1 being Morphism of b1,c1 st
  f = f1 holds G.f = Morph-Map(F,b1,c1).f1;
  G is surjective by A3;
  then G is onto;
  then
A6: G is coreflexive by FUNCTOR0:47;
  thus B2 is subcategory of A2 & B1 is subcategory of A1;
  consider H being Functor of B2,B1 such that
A7: H = G" and
A8: H is bijective contravariant by A3,FUNCTOR0:49;
  reconsider H as contravariant Functor of B2,B1 by A8;
  take H;
  thus H is bijective by A8;
A9: the carrier of B1 c= the carrier of A1 by ALTCAT_2:def 11;
  thus
A10: now
    let a be Object of B2, a1 be Object of A2;
    reconsider Ha = H.a as Object of A1 by A9;
    G.(H.a) = F.Ha by A4;
    then
A11: F.Ha = a by A3,A7,A6,Th1;
    assume a = a1;
    hence H.a = F".a1 by A1,A2,A11,Th1;
  end;
  let b,c be Object of B2, b1,c1 be Object of A2 such that
A12: <^b,c^> <> {} and
A13: b = b1 & c = c1;
  let f be Morphism of b,c, f1 be Morphism of b1,c1 such that
A14: f = f1;
A15: <^b,c^> c= <^b1,c1^> & f in <^b,c^> by A12,A13,ALTCAT_2:31;
A16: G.(H.b) = b & G.(H.c) = c by A3,A7,A6,Th1;
A17: <^H.c, H.b^> <> {} by A12,FUNCTOR0:def 19;
  then
A18: H.f in <^H.c, H.b^>;
A19: F.(F".b1) = b1 & F.(F".c1) = c1 by A1,A2,Th1;
A20: H.b = F".b1 & H.c = F".c1 by A10,A13;
  then
A21: <^H.c, H.b^> c= <^F".c1, F".b1^> by ALTCAT_2:31;
  then reconsider Hf = H.f as Morphism of F".c1, F".b1 by A18;
  G.(H.f) = Morph-Map(F,F".c1, F".b1).Hf by A5,A20,A17
    .= F.Hf by A21,A18,A15,A19,FUNCTOR0:def 16;
  then F.Hf = f by A3,A7,A17,A16,Th3;
  hence H.f = F9.f1 by A1,A14,A21,A18,A19,Th3
    .= Morph-Map(F",b1,c1).f1 by A21,A18,A15,FUNCTOR0:def 16;
end;
