
theorem
  for C be Category, I being Indexing of C for T being TargetCat of I
  for D,E being Categorial Category, F being Functor of T,D for G being Functor
  of D,E holds (G*F)*I = G*(F*I)
proof
  let C be Category;
  let I be Indexing of C;
  let T be TargetCat of I;
  let D,E be Categorial Category;
  let F be Functor of T,D;
  reconsider D9 = D as TargetCat of F*I by Th29;
  let G be Functor of D,E;
  reconsider G9 = G as Functor of D9, E;
  F*I = (F*(I-functor(C,T)))-indexing_of C by Def17;
  then
A1: (F*I)-functor(C,D9) = F*(I-functor(C,T)) by Th18;
  thus (G*F)*I = ((G*F)*(I-functor(C,T)))-indexing_of C by Def17
    .= (G9*((F*I)-functor(C,D9)))-indexing_of C by A1,RELAT_1:36
    .= G*(F*I) by Def17;
end;
