
theorem Th20:
  for C being Category, I being (Indexing of C), T being TargetCat
  of I holds pr2 (I-functor(C,T)) = I`2
proof
  let C be Category, I be Indexing of C;
  let T be TargetCat of I;
A1: dom (I-functor(C,T)) = 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;
    (I-functor(C,T)).f = [[I`1.dom f, I`1.cod f], I`2.f] by Def11;
    then ((I-functor(C,T)).x)`2 = I`2.f;
    hence (pr2 (I-functor(C,T))).x = I`2.x by A1,MCART_1:def 13;
  end;
  dom pr2 (I-functor(C,T)) = dom (I-functor(C,T)) & dom I`2 = the carrier'
  of C by MCART_1:def 13,PARTFUN1:def 2;
  hence thesis by A1,A2;
end;
