
theorem
  for C being composable with_identities CategoryStr,
      a,b being Object of C, f being Morphism of a,b
  holds Hom(a,b) <> {} & b is initial implies f is epimorphism
proof
  let C be composable with_identities CategoryStr,
      a,b be Object of C, f be Morphism of a,b;
  assume that
A1: Hom(a,b) <> {} and
A2: b is initial;
  now
    let c be Object of C such that Hom(b,c)<>{};
    let g,h be Morphism of b,c such that g*f=h*f;
    consider f1 be Morphism of b,c such that
A3: for g1 being Morphism of b,c holds f1=g1 by A2;
    f1 = g by A3;
    hence g=h by A3;
  end;
  hence thesis by A1,CAT_7:def 6;
end;
