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 Th15:
  for F being Functor of [:A,B:],C, f being (Morphism of A), b
  being Object of B holds (F?-f).b = F.(f, id b)
proof
  let F be Functor of [:A,B:],C, f be (Morphism of A), b be Object of B;
  reconsider
  G = F as Function of [:the carrier' of A,the carrier' of B:], the
  carrier' of C;
  reconsider Ff = (curry G).f as Function of the carrier' of B,the carrier' of
  C;
A1: id b = (IdMap B).b by ISOCAT_1:def 12;
  F?-dom f is_naturally_transformable_to F?-cod f by Th14;
  then F?-dom f is_transformable_to F?-cod f;
  hence (F?-f).b = (curry(F,f)*IdMap B).b by NATTRA_1:def 5
    .= Ff.(id b qua Morphism of B) by A1,FUNCT_2:15
    .= F.(f,id b) by FUNCT_5:69;
end;
