reserve A,B,C for Category,
  F,F1,F2,F3 for Functor of A,B,
  G for Functor of B, C;
reserve m,o for set;
reserve t for natural_transformation of F,F1,
  t1 for natural_transformation of F1,F2;

theorem Th28:
  F1 is_naturally_transformable_to F2 iff [[F1,F2],t1] in NatTrans (A,B)
proof
  thus F1 is_naturally_transformable_to F2 implies [[F1,F2],t1] in NatTrans(A,
  B) by Def15;
  assume [[F1,F2],t1] in NatTrans(A,B);
  then consider
  F19,F29 being Functor of A,B, t being natural_transformation of F19
  ,F29 such that
A1: [[F1,F2],t1] = [[F19,F29],t] and
A2: F19 is_naturally_transformable_to F29 by Def15;
A3: [F1,F2] = [F19,F29] by A1,XTUPLE_0:1;
  then F1 = F19 by XTUPLE_0:1;
  hence thesis by A2,A3,XTUPLE_0:1;
end;
