
theorem
  for C1,C2 being category holds Functors(C1,C2) ~= C2|^C1
  proof
    let C1,C2 be category;
    per cases;
    suppose
      C1 is empty;
      hence thesis by Th28;
    end;
    suppose
      C2 is empty & C1 is non empty;
      hence thesis by CAT_7:13;
    end;
    suppose
A1:   C1 is non empty & C2 is non empty;
      set C = Functors(C1,C2);
      consider E be Functor of C [x] C1,C2 such that
A2:   E is covariant and
A3:   for D being category,
      F being Functor of D [x] C1,C2 st F is covariant holds
      ex H being Functor of D, C st
      H is covariant & F = E (*) (H [x] id C1) &
      for H1 being Functor of D, C st
      H1 is covariant & F = E (*) (H1 [x] id C1)
      holds H = H1 by A1,Lm6;
A4:  C,E is_exponent_of C1,C2 by A2,A3,Def34;
      C2|^C1,eval(C1,C2) is_exponent_of C1,C2 by Th72;
      hence thesis by A2,A4,Th73;
    end;
  end;
