reserve C for category,
  o1, o2, o3 for Object of C;

theorem Th4:
  for C being with_units non empty AltCatStr, o being Object of C
  holds idm o is epi mono
proof
  let C be with_units non empty AltCatStr, o be Object of C;
  thus idm o is epi
  proof
    let o1 be Object of C such that
A1: <^o,o1^> <> {};
    let B, C be Morphism of o, o1 such that
A2: B * idm o = C * idm o;
    thus B = B * idm o by A1,ALTCAT_1:def 17
      .= C by A1,A2,ALTCAT_1:def 17;
  end;
  let o1 be Object of C such that
A3: <^o1,o^> <> {};
  let B, C be Morphism of o1, o such that
A4: idm o * B = idm o * C;
  thus B = idm o * B by A3,ALTCAT_1:20
    .= C by A3,A4,ALTCAT_1:20;
end;
