reserve B,C,D for Category;

theorem Th4:
  for a,b being Object of C
   holds Hom(a,b) = Hom(b opp,a opp)
proof
 let a,b be Object of C;
 thus Hom(a,b) c= Hom(b opp,a opp)
  proof let x be object;
   assume
A1:  x in Hom(a,b);
    then reconsider f = x as Morphism of C;
    reconsider g = f as Morphism of C opp;
    dom f = a & cod f = b by A1,CAT_1:1;
    then dom g = b opp & cod g = a opp;
   hence x in Hom(b opp,a opp);
  end;
 let x be object;
 assume
A2: x in Hom(b opp,a opp);
  then reconsider f = x as Morphism of C opp;
  reconsider g = f as Morphism of C;
  dom f = b opp & cod f = a opp by A2,CAT_1:1;
  then dom g = a & cod g = b;
 hence x in Hom(a,b);
end;
