
theorem Th18:
  for C being Category, D being Categorial Category, F being
  Functor of C,D for T being TargetCat of F-indexing_of C holds F = (F
  -indexing_of C)-functor(C,T)
proof
  let C be Category, D be Categorial Category, F be Functor of C,D;
  set I = F-indexing_of C;
  let T be TargetCat of I;
A1: dom F = the carrier' of C by FUNCT_2:def 1;
A2: now
    let x be object;
    assume x in the carrier' of C;
    then reconsider f = x as Morphism of C;
    set h = F.f;
A3: dom h = (Obj F).dom f & cod h = (Obj F).cod f by CAT_1:69;
A4: dom h = h`11 & cod h = h`12 by CAT_5:13;
    then consider g being Functor of h`11, h`12 such that
A5: h = [[h`11, h`12], g] by CAT_5:def 6;
    I`2.f = h`2 by A1,MCART_1:def 13
      .= g by A5;
    hence F.x = (I-functor(C,T)).x by A4,A5,A3,Def11;
  end;
  thus thesis by A2;
end;
