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 Th20:
  for F1,F2 being Functor of [:A,B:],C holds export F1 = export F2
  implies F1 = F2
proof
  let F1,F2 be Functor of [:A,B:],C such that
A1: export F1 = export F2;
  now
    let fg be Morphism of [:A,B:];
    consider f being (Morphism of A), g being Morphism of B such that
A2: fg = [f,g] by CAT_2:27;
A3: dom id cod f = cod f & dom g = cod id dom g;
A4: fg = [(id cod f)(*)f,g] by A2,Th3
      .= [(id cod f)(*)f,g(*)(id dom g)] by Th4
      .= [id cod f,g](*)[f, id dom g] by A3,CAT_2:29;
A5: [[F1?-dom f,F1?-cod f],F1?-f] = (export F2).f by A1,Def4
      .= [[F2?-dom f,F2?-cod f],F2?-f] by Def4;
    then
A6: [F1?-dom f,F1?-cod f] = [F2?-dom f,F2?-cod f] by XTUPLE_0:1;
    then
A7: F1?-dom f = F2?-dom f by XTUPLE_0:1;
A8: F1?-cod f = F2?-cod f by A6,XTUPLE_0:1;
    then
A9: F1.(id cod f, g) = (F2?-cod f).g by CAT_2:36
      .= F2.(id cod f, g) by CAT_2:36;
A10: cod[f,id dom g] = [cod f, cod id dom g] by CAT_2:28
      .= [cod f, dom g]
      .= [dom id cod f, dom g]
      .= dom[id cod f,g] by CAT_2:28;
    F1.(f, id dom g) = (F1?-f).dom g by Th15
      .= (F2?-f).dom g by A5,A7,A8,XTUPLE_0:1
      .= F2.(f,id dom g) by Th15;
    hence F1.fg = (F2.[id cod f,g])(*)(F2.[f, id dom g]) by A4,A9,A10,CAT_1:64
      .= F2.fg by A4,A10,CAT_1:64;
  end;
  hence thesis by FUNCT_2:63;
end;
