
theorem Th65:
  for C1,C2 being non empty category, f being morphism of Functors(C1,C2) holds
  ex F1,F2 being covariant Functor of C1,C2,
     T being natural_transformation of F1,F2 st f = [[F1,F2],T] &
     dom f = [[F1,F1],F1] & cod f = [[F2,F2],F2]
  proof
    let C1,C2 be non empty category;
    set C = Functors(C1,C2);
    let f be morphism of C;
    consider f1 be morphism of C such that
A1: dom f = f1 & f |> f1 & f1 is identity by CAT_6:def 18;
    consider f2 be morphism of C such that
A2: cod f = f2 & f2 |> f & f2 is identity by CAT_6:def 19;
    consider G1,G11,G12 be covariant Functor of C1,C2,
    T11 be natural_transformation of G11,G1,
    T12 be natural_transformation of G1,G12 such that
A3: f = [[G1,G12],T12] & f1 = [[G11,G1],T11] & f(*)f1 = [[G11,G12],T12`*`T11] &
    for g1,g2 being morphism of C1 st g2 |> g1 holds T12.g2 |> T11.g1 &
    (T12`*`T11).(g2(*)g1) = (T12.g2)(*)(T11.g1) by A1,Th63;
    consider F1 be covariant Functor of C1,C2 such that
A4: f1 = [[F1,F1],F1] by A1,Th64;
    [G11,G1] = [F1,F1] by A3,A4,XTUPLE_0:1;
    then
A5: G1 = F1 by XTUPLE_0:1;
    consider G2,G21,G22 be covariant Functor of C1,C2,
    T21 be natural_transformation of G21,G2,
    T22 be natural_transformation of G2,G22 such that
A6: f2 = [[G2,G22],T22] & f = [[G21,G2],T21] & f2(*)f = [[G21,G22],T22`*`T21] &
    for g1,g2 being morphism of C1 st g2 |> g1 holds T22.g2 |> T21.g1 &
    (T22`*`T21).(g2(*)g1) = (T22.g2)(*)(T21.g1) by A2,Th63;
    consider F2 be covariant Functor of C1,C2 such that
A7: f2 = [[F2,F2],F2] by A2,Th64;
    [G2,G22] = [F2,F2] by A6,A7,XTUPLE_0:1;
    then
A8: G2 = F2 by XTUPLE_0:1;
A9: [G1,G12] = [G21,G2] by A3,A6,XTUPLE_0:1;
    then reconsider T = T12 as natural_transformation of F1,F2
    by A5,A8,XTUPLE_0:1;
    take F1,F2,T;
    thus f = [[F1,F2],T] by A3,A5,A9,A8,XTUPLE_0:1;
    thus dom f = [[F1,F1],F1] by A1,A4;
    thus cod f = [[F2,F2],F2] by A2,A7;
  end;
