reserve x,y for set;

theorem Th2:
  for A,B being transitive with_units non empty AltCatStr for F
  being Covariant feasible FunctorStr over A,B for G being Covariant 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.a1, F.a2
  holds F.f = g iff G.g = f
proof
  let A,B be transitive with_units non empty AltCatStr;
  let F be Covariant feasible FunctorStr over A,B;
  let G be Covariant 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.a1,F.a2^> <> {} by A3,FUNCTOR0:def 18;
  F is surjective by A1;
  then F is onto;
  then F is reflexive coreflexive by FUNCTOR0:44;
  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.a1, F.a2;
  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,FUNCTOR3:6;
  F * G = id B by A1,A2,FUNCTOR1:18;
  then g = (F * G).g by A4,FUNCTOR0:31;
  hence thesis by A4,A5,FUNCTOR3:6;
end;
