reserve x,y for set;

theorem Th4:
  for A,B being category, F being Functor of A,B st F is bijective
  for G being Functor of B,A st F*G = id B holds the FunctorStr of G = F"
proof
  let A,B be category, F be Functor of A,B;
  assume
A1: F is bijective;
  then reconsider FF = F" as feasible FunctorStr over B,A by FUNCTOR0:35;
A2: F"*F = id A by A1,FUNCTOR1:19;
  let G be Functor of B,A;
  assume F * G = id B;
  then (id A)*G = FF * id B by A2,FUNCTOR0:32
    .= F" by FUNCTOR3:5;
  hence thesis by FUNCTOR3:4;
end;
