
theorem Th37:
  for C,D being Category, F being Functor of C,D, I being Indexing
of D for T being TargetCat of I, J being Indexing of T holds (J*I)*F = J*(I*F)
proof
  let C,D be Category, F be Functor of C,D, I be Indexing of D;
  let T be TargetCat of I, J be Indexing of T;
A1: I*F = ((I-functor(D,T))*F)-indexing_of C & Image ((I-functor(D,T))*F) is
  Subcategory of T by Th23;
  T is TargetCat of I*F by Th24;
  then rng (I*F) is Subcategory of T by Th14;
  then
A2: Image ((I*F)-functor(C, rng (I*F))) is Subcategory of T by CAT_5:4;
  thus (J*I)*F = (J*(I-functor(D,T)))*F by Th32
    .= J*((I-functor(D,T))*F) by Th26
    .= J*(I*F) by A1,A2,Th18,Th22;
end;
