
theorem Th48:
  for A,B being transitive with_units non empty AltCatStr,
  F being covariant Functor of A,B st F is bijective
  ex G being Functor of B,A st G = F" & G is bijective covariant
proof
  let A,B be transitive with_units non empty AltCatStr,
  F be covariant Functor of A,B;
  assume
A1: F is bijective;
  then F is injective;
  then F is one-to-one;
  then
A2: the ObjectMap of F is one-to-one;
A3: F is feasible by Def25;
  then
A4: F" is bijective feasible by A1,Th35;
A5: F is id-preserving by Def25;
A6: F is comp-preserving by Def26;
  F is surjective by A1;
  then
A7: F is onto;
  then
A8: the ObjectMap of F is onto;
A9: F is Covariant by Def26;
A10: F is coreflexive by A7,Th46;
A11: F" is Covariant proof F is Covariant by Def26;
    then the ObjectMap of F is Covariant;
    then consider f being Function of the carrier of A, the carrier of B
    such that
A12: the ObjectMap of F = [:f,f:];
A13: f is one-to-one by A2,A12,Th7;
    then
A14: dom(f") = rng f by FUNCT_1:33;
A15: rng(f") = dom f by A13,FUNCT_1:33;
A16: rng[:f,f:] = [:the carrier of B,the carrier of B:]
    by A8,A12;
    rng[:f,f:] = [:rng f,rng f:] by FUNCT_3:67;
    then rng f = the carrier of B by A16,ZFMISC_1:92;
    then reconsider g = f" as Function of the carrier of B, the carrier of A
    by A14,A15,FUNCT_2:def 1,RELSET_1:4;
    take g;
    [:f,f:]" = [:g,g:] by A13,Th6;
    hence thesis by A1,A12,Def38;
  end;
A17: F" is id-preserving by A1,A3,A5,A10,Th37;
  F" is comp-preserving by A1,A3,A6,A9,A10,Th42;
  then reconsider G = F" as Functor of B,A by A4,A11,A17,Def25;
  take G;
  thus G = F";
  thus G is bijective by A1,A3,Th35;
  thus G is Covariant by A11;
  thus thesis by A1,A3,A6,A9,A10,Th42;
end;
