reserve e for set;

theorem Th15:
  for C being Category holds the_comps_of C is with_left_units with_right_units
proof
  let C be Category;
  thus the_comps_of C is with_left_units
  proof
    let i be Object of C;
    take id i;
    thus id i in (the_hom_sets_of C).(i,i) by Th12;
    let j be Object of C, f be set;
    assume f in (the_hom_sets_of C).(j,i);
    then
A1: f in Hom(j,i) by Def3;
    then reconsider m = f as Morphism of j,i by CAT_1:def 5;
    Hom(i,i) <> {};
    hence (the_comps_of C).(j,i,i).(id i,f) = (id i)*m by A1,Th13
      .= f by A1,CAT_1:28;
  end;
  let j be Object of C;
  take id j;
  thus id j in (the_hom_sets_of C).(j,j) by Th12;
  let i be Object of C, f be set;
  assume f in (the_hom_sets_of C).(j,i);
  then
A2: f in Hom(j,i) by Def3;
  then reconsider m = f as Morphism of j,i by CAT_1:def 5;
  Hom(j,j) <> {};
  hence (the_comps_of C).(j,j,i).(f,id j) = m*(id j) by A2,Th13
    .= f by A2,CAT_1:29;
end;
