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 Th31:
  for t9 being natural_transformation of F2,F3 for f,g being
  Morphism of Functors(A,B) st f = [[F,F1],t] & g = [[F2,F3],t9] holds [g,f] in
  dom the Comp of Functors(A,B) iff F1 = F2
proof
  let t9 be natural_transformation of F2,F3;
  let f,g be Morphism of Functors(A,B);
  assume that
A1: f = [[F,F1],t] and
A2: g = [[F2,F3],t9];
  thus [g,f] in dom the Comp of Functors(A,B) implies F1 = F2
  proof
    assume [g,f] in dom the Comp of Functors(A,B);
    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
A3: f = [[F9,F19],t9] and
A4: g = [[F19,F29],t19] and
    (the Comp of Functors(A,B)).[g,f] = [[F9,F29],t19`*`t9] by Def16;
    thus F1 = [[F,F1],t]`1`2
      .= [[F9,F19],t9]`1`2 by A1,A3
      .= [F9,F19]`2
      .= [[F19,F29],t19]`1`1
      .= [F2,F3]`1 by A2,A4
      .= F2;
  end;
A5: cod f = F1 by A1,Th29;
  dom g = F2 by A2,Th29;
  hence thesis by A5,Def16;
end;
