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 Th16:
  for F being Functor of [:A,B:],C, a being Object of A holds id(F
  ?-a) = F?-id a
proof
  let F be Functor of [:A,B:],C, a be Object of A;
  reconsider
  G = F as Function of [:the carrier' of A,the carrier' of B:], the
  carrier' of C;
  reconsider Ff = (curry G).(id a qua Morphism of A) as Function of the
  carrier' of B,the carrier' of C;
  reconsider I = F?-id a as transformation of F?-a,F?-a;
  now
    let b be Object of B;
A1: id b = (IdMap B).b by ISOCAT_1:def 12;
    thus (I qua Function of the carrier of B, the carrier' of C).b = Ff.(id b
    qua Morphism of B) by A1,FUNCT_2:15
      .= F.(id a,id b) by FUNCT_5:69
      .= 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 Th8;
  end;
  hence thesis by NATTRA_1:def 4;
end;
