reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;
reserve a,b for Object of Ens(V);
reserve f,g,f1,f2 for Morphism of Ens(V);
reserve C for Category,
  a,b,a9,b9,c for Object of C,
  f,g,h,f9,g9 for Morphism of C;

theorem
  Hom(C) c= V implies (Obj (hom?-(V,a))).b = Hom(a,b)
proof
  assume
A1: Hom(C) c= V;
  Hom(a,b) in Hom(C);
  then reconsider A = Hom(a,b) as Element of V by A1;
  set d = @A;
  (hom?-(V,a)).(id b) = (hom?-(a)).(id b) by A1,Def24
    .= id d by A1,Lm9;
  hence thesis by CAT_1:67;
end;
