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 Th22:
  for F being Functor of [:A,B:],C holds id export F = export id F
proof
  let F be Functor of [:A,B:],C;
  reconsider
  s = id F as Function of [:the carrier of A, the carrier of B:],
  the carrier' of C;
  now
    let a be Object of A;
    reconsider ff = (export F).a as Functor of B,C by Th5;
A1: now
      let b be Object of B;
      thus (id ff qua Function of the carrier of B, the carrier' of C).b = (id
      ff).b by NATTRA_1:def 5
        .= id(ff.b) by NATTRA_1:20
        .= (ff qua Function of the carrier' of B, the carrier' of C) .id b
      by CAT_1:71
        .= (F?-a).id b by Th18
        .= F.(id a, id b) by CAT_2:36
        .= F.(id[a,b] qua Morphism of [:A,B:]) by CAT_2:31
        .= id(F.[a,b]) by CAT_1:71
        .= (id F).[a,b] by NATTRA_1:20
        .= (id F qua Function of the carrier of [:A,B:], the carrier' of C)
      .(a,b) by NATTRA_1:def 5
        .= (curry s).a.b by FUNCT_5:69;
    end;
    thus (id export F).a = id((export F).a) by NATTRA_1:20
      .= [[ff,ff],id ff] by NATTRA_1:def 17
      .= [[(export F).a, (export F).a],(curry s).a] by A1,FUNCT_2:63
      .= (export id F).a by Def5;
  end;
  hence thesis by NATTRA_1:19;
end;
