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

theorem Th24:
  for F1,F2 being Functor of [:A,B:],C st F1
is_naturally_transformable_to F2 for t1,t2 being natural_transformation of F1,
  F2 holds export t1 = export t2 implies t1 = t2
proof
  let F1,F2 be Functor of [:A,B:],C;
  assume
A1: F1 is_naturally_transformable_to F2;
  then
A2: F1 is_transformable_to F2;
  let t1,t2 be natural_transformation of F1,F2 such that
A3: export t1 = export t2;
  now
    reconsider
    s1 = t1, s2 = t2 as Function of [:the carrier of A, the carrier
    of B:], the carrier' of C;
    let ab be Object of [:A,B:];
    consider a being Object of A, b being Object of B such that
A4: ab = [a,b] by DOMAIN_1:1;
    [[(export F1).a,(export F2).a],(curry s1).a] = (export t1).a by A1,Def5
      .= [[(export F1).a,(export F2).a],(curry s2).a] by A1,A3,Def5;
    then
A5: (curry s1).a = (curry s2.a) by XTUPLE_0:1;
    thus t1.ab = s1.(a,b) by A2,A4,NATTRA_1:def 5
      .= ((curry s2).a).b by A5,FUNCT_5:69
      .= s2.(a,b) by FUNCT_5:69
      .= t2.ab by A2,A4,NATTRA_1:def 5;
  end;
  hence thesis by A1,ISOCAT_1:26;
end;
