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 Th11:
  for F1,F2 being Functor of [:A,B:],C st F1
is_naturally_transformable_to F2 for t being natural_transformation of F1,F2, a
being Object of A holds F1?-a is_naturally_transformable_to F2?-a & (curry t).a
  is natural_transformation of F1?-a,F2?-a
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 u be natural_transformation of F1,F2, a be Object of A;
  reconsider
  t=u as Function of [:the carrier of A, the carrier of B:], the
  carrier' of C;
A3: F1?-a is_transformable_to F2?-a
  proof
    let b be Object of B;
    (F1?-a).b = F1.[a,b] & (F2?-a).b = F2.[a,b] by Th8;
    hence thesis by A2;
  end;
A4: now
    let b be Object of B;
A5: (F1?-a).b = F1.[a,b] & (F2?-a).b = F2.[a,b] by Th8;
A6: Hom((F1?-a).b,(F2?-a).b) <> {} by A3;
    (curry t).a.b = t.(a,b) by FUNCT_5:69
      .= u.[a,b] by A2,NATTRA_1:def 5;
    hence (curry t).a.b in Hom((F1?-a).b,(F2?-a).b) by A5,A6,CAT_1:def 5;
  end;
  now
    let b be Object of B;
    (curry t).a.b in Hom((F1?-a).b,(F2?-a).b) by A4;
    hence (curry t).a.b is Morphism of (F1?-a).b,(F2?-a).b by CAT_1:def 5;
  end;
  then reconsider s = (curry t).a as transformation of F1?-a,F2?-a by A3,
NATTRA_1:def 3;
A7: now
    let b1,b2 be Object of B such that
A8: Hom(b1,b2) <> {};
A9: Hom((F1?-a).b1,(F1?-a).b2) <> {} by A8,CAT_1:84;
    let f be Morphism of b1,b2;
A10: Hom(a,a) <> {};
    then reconsider g = [id a,f] as Morphism of [a,b1],[a,b2] by A8,CAT_2:33;
A11: Hom([a,b1],[a,b2]) <> {} by A8,A10,Th9;
    then
A12: Hom(F1.[a,b1],F1.[a,b2]) <> {} by CAT_1:84;
A13: s.b1 = (curry t).a.b1 by A3,NATTRA_1:def 5
      .= t.(a,b1) by FUNCT_5:69
      .= u.[a,b1] by A2,NATTRA_1:def 5;
A14: Hom(F1.[a,b2],F2.[a,b2]) <> {} by A2;
A15: Hom((F2?-a).b1,(F2?-a).b2) <> {} by A8,CAT_1:84;
A16: (F1?-a).b1 = F1.[a,b1] & (F2?-a).b1 = F2.[a,b1] by Th8;
A17: (F1?-a).b2 = F1.[a,b2] & (F2?-a).b2 = F2.[a,b2] by Th8;
A18: Hom(F1.[a,b1],F2.[a,b1]) <> {} by A2;
A19: Hom(F2.[a,b1],F2.[a,b2]) <> {} by A11,CAT_1:84;
    s.b2 = (curry t).a.b2 by A3,NATTRA_1:def 5
      .= t.(a,b2) by FUNCT_5:69
      .= u.[a,b2] by A2,NATTRA_1:def 5;
    hence s.b2*(F1?-a)/.f
       = (u.[a,b2])(*)((F1?-a)/.f) by A14,A9,A17,CAT_1:def 13
      .= (u.[a,b2])(*)((F1?-a).(f qua Morphism of B)) by A8,CAT_3:def 10
      .= (u.[a,b2])(*)(F1.(id a,f)) by CAT_2:36
      .= (u.[a,b2])(*)(F1/.g qua Morphism of C) by A11,CAT_3:def 10
      .= u.[a,b2]*F1/.g by A12,A14,CAT_1:def 13
      .= F2/.g*u.[a,b1] by A1,A11,NATTRA_1:def 8
      .= (F2/.g qua Morphism of C)(*)(u.[a,b1]) by A18,A19,CAT_1:def 13
      .= F2.(id a,f)(*)(u.[a,b1]) by A11,CAT_3:def 10
      .= ((F2?-a).(f qua Morphism of B))(*)(u.[a,b1]) by CAT_2:36
      .= ((F2?-a)/.f qua Morphism of C)(*)(s.b1) by A8,A13,CAT_3:def 10
      .= (F2?-a)/.f*s.b1 by A18,A15,A16,CAT_1:def 13;
  end;
  hence F1?-a is_naturally_transformable_to F2?-a by A3;
  hence thesis by A7,NATTRA_1:def 8;
end;
