
theorem Th11:
  for C being Category, I being Indexing of C for T1,T2 being
TargetCat of I holds I-functor(C,T1) = I-functor(C,T2) & Obj (I-functor(C,T1))
  = Obj (I-functor(C,T2))
proof
  let C be Category, I be Indexing of C;
  let T1,T2 be TargetCat of I;
A1: now
    let x be object;
    assume x in the carrier' of C;
    then reconsider f = x as Morphism of C;
    thus (I-functor(C,T1)).x = [[I`1.dom f, I`1.cod f], I`2.f] by Def11
      .= (I-functor(C,T2)).x by Def11;
  end;
  thus I-functor(C,T1) = I-functor(C,T2) by A1;
A2: now
    let x be object;
    assume x in the carrier of C;
    then reconsider a = x as Object of C;
    thus (Obj (I-functor(C,T1))).x = I`1.a by Lm3
      .= (Obj (I-functor(C,T2))).x by Lm3;
  end;
  thus thesis by A2;
end;
