reserve B,C,D for Category;

theorem Th9:
  for a,b being Object of C st Hom(a,b)<>{}
  for f being Morphism of a,b holds dom(f opp) = cod f & cod(f opp) = dom f
 proof let a,b be Object of C;
  assume
A1:  Hom(a,b)<>{};
   then
A2:  Hom(b opp,a opp)<>{} by Th4;
  let f be Morphism of a,b;
  thus dom(f opp) = b by A2,CAT_1:5
    .= cod f by A1,CAT_1:5;
  thus cod(f opp) = a by A2,CAT_1:5
    .= dom f by A1,CAT_1:5;
 end;
