reserve x,y for set;

theorem Th3:
  for A,B being transitive with_units non empty AltCatStr for F
  being Contravariant feasible FunctorStr over A,B for G being Contravariant
feasible FunctorStr over B,A st F is bijective & G = F" for a1,a2 being Object
of A st <^a1,a2^> <> {} for f being Morphism of a1,a2, g being Morphism of F.a2
  , F.a1 holds F.f = g iff G.g = f
proof
  let A,B be transitive with_units non empty AltCatStr;
  let F be Contravariant feasible FunctorStr over A,B;
  let G be Contravariant feasible FunctorStr over B,A such that
A1: F is bijective and
A2: G = F";
  let a1,a2 be Object of A such that
A3: <^a1,a2^> <> {};
A4: <^F.a2,F.a1^> <> {} by A3,FUNCTOR0:def 19;
  F is surjective by A1;
  then F is onto;
  then F is reflexive coreflexive by FUNCTOR0:45;
  then
A5: G.(F.a1) = a1 & G.(F.a2) = a2 by A1,A2,Th1;
  let f be Morphism of a1,a2, g be Morphism of F.a2, F.a1;
  F" * F = id A by A1,FUNCTOR1:19;
  then f = (G * F).f by A2,A3,FUNCTOR0:31;
  hence F.f = g implies G.g = f by A3,FUNCTOR3:7;
  F * G = id B by A1,A2,FUNCTOR1:18;
  then g = (F * G).g by A4,FUNCTOR0:31;
  hence thesis by A4,A5,FUNCTOR3:7;
end;
