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 Th13:
  for F being Functor of [:A,B:], C for a,b being Object of A st
  Hom(a,b) <> {} for f being Morphism of a,b holds F?-a
  is_naturally_transformable_to F?-b & curry(F,f)*IdMap B is
  natural_transformation of F?-a,F?-b
proof
  let F be Functor of [:A,B:], C;
  let a1,a2 be Object of A such that
A1: Hom(a1,a2) <> {};
  reconsider
  G = F as Function of [:the carrier' of A,the carrier' of B:], the
  carrier' of C;
  let f be Morphism of a1,a2;
  reconsider Ff = (curry G).(f qua Morphism of A) as Function of the carrier'
  of B,the carrier' of C;
A2: now
    let b be Object of B;
    f in Hom(a1,a2) & id b in Hom(b,b) by A1,CAT_1:def 5;
    then [f,id b] in Hom([a1,b],[a2,b]) by Th12;
    then
A3: F.(f,id b) in Hom(F.[a1,b],F.[a2,b]) by CAT_1:81;
A4: id b = (IdMap B).b by ISOCAT_1:def 12;
    (F?-a1).b = F.[a1,b] & (F?-a2).b = F.[a2,b] by Th8;
    then Ff.(id b qua Morphism of B) in Hom((F?-a1).b,(F?-a2).b)
         by A3,FUNCT_5:69;
    hence (curry(F,f)*IdMap B).b in Hom((F?-a1).b,(F?-a2).b)
          by A4,FUNCT_2:15;
  end;
A5: F?-a1 is_transformable_to F?-a2
  by A2;
  reconsider FfI = curry(F,f)*IdMap B as Function of the carrier of B, the
  carrier' of C;
  now
    let b be Object of B;
    (curry(F,f)*IdMap B).b in Hom((F?-a1).b,(F?-a2).b) by A2;
    hence FfI.b is Morphism of (F?-a1).b,(F?-a2).b by CAT_1:def 5;
  end;
  then reconsider
  t = curry(F,f)*IdMap B as transformation of F?-a1,F?-a2 by A5,NATTRA_1:def 3;
A6: now
    reconsider ida1 = id a1, ida2 = id a2 as Morphism of A;
A7: Hom(a1,a1) <> {};
A8: Hom(a2,a2) <> {};
    let b1,b2 be Object of B such that
A9: Hom(b1,b2) <> {};
A10: Hom((F?-a2).b1,(F?-a2).b2) <> {} by A9,CAT_1:84;
    let g be Morphism of b1,b2;
    reconsider idb1 = id b1, idb2 = id b2 as Morphism of B;
A11: Hom(b1,b1) <> {};
A12: dom id b2 = b2
      .= cod g by A9,CAT_1:5;
    Hom(b2,b2) <> {};
    then
A13: [f(*)ida1,idb2(*)g] = [f(*)ida1,(id b2)*g] by A9,CAT_1:def 13
      .= [f(*)ida1,g] by A9,CAT_1:28
      .= [f*id a1,g] by A1,A7,CAT_1:def 13
      .= [f,g] by A1,CAT_1:29
      .= [id a2*f,g] by A1,CAT_1:28
      .= [ida2(*)f,g] by A1,A8,CAT_1:def 13
      .= [ida2(*)f,g*id b1] by A9,CAT_1:29
      .= [ida2(*)f,g(*)idb1] by A9,A11,CAT_1:def 13;
A14: Hom((F?-a1).b2,(F?-a2).b2) <> {} & t.b2 = FfI.b2 by A5,NATTRA_1:def 5;
A15: cod id a1 = a1
      .= dom f by A1,CAT_1:5;
    then
A16: cod[id a1,g] = [dom f,cod g] by CAT_2:28
      .= dom[f,id b2] by A12,CAT_2:28;
A17: Hom((F?-a1).b1,(F?-a2).b1) <> {} & t.b1 = FfI.b1 by A5,NATTRA_1:def 5;
A18: dom g = b1 by A9,CAT_1:5
      .= cod id b1;
A19: dom id a2 = a2
      .= cod f by A1,CAT_1:5;
    then
A20: dom[id a2,g] = [cod f,dom g] by CAT_2:28
      .= cod[f,id b1] by A18,CAT_2:28;
A21: id b2 = (IdMap B).b2 by ISOCAT_1:def 12;
A22: id b1 = (IdMap B).b1 by ISOCAT_1:def 12;
    Hom((F?-a1).b1,(F?-a1).b2) <> {} by A9,CAT_1:84;
    hence t.b2*(F?-a1)/.g = (FfI.b2)(*)((F?-a1)/.g) by A14,CAT_1:def 13
      .= (Ff.(id b2 qua Morphism of B))(*)((F?-a1)/.g) by A21,FUNCT_2:15
      .= (F.(f,id b2))(*)((F?-a1)/.g) by FUNCT_5:69
      .= (F.[f,id b2])(*)((F?-a1).(g qua Morphism of B)) by A9,CAT_3:def 10
      .= (F.(f,id b2))(*)(F.(id a1,g)) by CAT_2:36
      .= F.([f,id b2](*)[id a1,g]) by A16,CAT_1:64
      .= F.[f(*)ida1,idb2(*)g] by A15,A12,CAT_2:29
      .= F.([ida2,g](*)[f,idb1]) by A19,A18,A13,CAT_2:29
      .= F.(id a2,g)(*)(F.[f,id b1]) by A20,CAT_1:64
      .= ((F?-a2).(g qua Morphism of B))(*)(F.[f,id b1]) by CAT_2:36
      .= ((F?-a2)/.g)(*)(F.(f,id b1)) by A9,CAT_3:def 10
      .= ((F?-a2)/.g)(*)(Ff.(id b1 qua Morphism of B)) by FUNCT_5:69
      .= ((F?-a2)/.g)(*)(FfI.b1) by A22,FUNCT_2:15
      .= (F?-a2)/.g*t.b1 by A10,A17,CAT_1:def 13;
  end;
  hence F?-a1 is_naturally_transformable_to F?-a2 by A5;
  hence thesis by A6,NATTRA_1:def 8;
end;
