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
  for f,g being Morphism of Functors(A,B) st f = [[F,F1],t] & g = [[F1,
  F2],t1] holds g(*)f = [[F,F2],t1`*`t]
proof
  let f,g be Morphism of Functors(A,B);
  assume that
A1: f = [[F,F1],t] and
A2: g = [[F1,F2],t1];
A3: [g,f] in dom the Comp of Functors(A,B) by A1,A2,Th31;
  then consider
  F9,F19,F29 being Functor of A,B, t9 being natural_transformation of
  F9,F19, t19 being natural_transformation of F19,F29 such that
A4: f = [[F9,F19],t9] and
A5: g = [[F19,F29],t19] and
A6: (the Comp of Functors(A,B)).(g,f) = [[F9,F29],t19`*`t9] by Def16;
A7: t1 = t19 by A2,A5,XTUPLE_0:1;
A8: [F9,F19] = [F,F1] by A1,A4,XTUPLE_0:1;
  then
A9: F = F9 by XTUPLE_0:1;
  [F19,F29] = [F1,F2] by A2,A5,XTUPLE_0:1;
  then
A10: F2 = F29 by XTUPLE_0:1;
A11: F1 = F19 by A8,XTUPLE_0:1;
  t = t9 by A1,A4,XTUPLE_0:1;
  hence thesis by A3,A6,A7,A9,A11,A10,CAT_1:def 1;
end;
