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,C))).[a opp,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;
A2: id(a opp) = id a by OPPCAT_1:71;
  set d = @A;
  (hom??(V,C)).(id[a opp,b]) = (hom??(V,C)).[id(a opp),id b] by CAT_2:31
    .= (hom??(C)).[id a,id b] by A1,Def26,A2
    .= id d by A1,Lm11;
  hence thesis by CAT_1:67;
end;
