
theorem
  for C being Category, I being Indexing of C for T being TargetCat of I
, D being Categorial Category for F being Functor of T,D, J being Indexing of D
  holds (J*F)*I = J*(F*I)
proof
  let C be Category, I be Indexing of C;
  let T be TargetCat of I;
  let D be Categorial Category, F be Functor of T,D;
  let J be Indexing of D;
A1: F*I = (F*(I-functor(C,T)))-indexing_of C & Image (F*(I-functor(C,T))) is
  Subcategory of D by Def17;
  D is TargetCat of F*I by Th29;
  then rng (F*I) is Subcategory of D by Th14;
  then
A2: Image ((F*I)-functor(C, rng (F*I))) is Subcategory of D by CAT_5:4;
  thus (J*F)*I = (J*F)*(I-functor(C,T)) by Th32
    .= J*(F*(I-functor(C,T))) by Th26
    .= J*(F*I) by A1,A2,Th18,Th22;
end;
