
theorem
  for A, B being category st A, B are_opposite
  for a, b being Object of A st <^a,b^> <> {} & <^b,a^> <> {}
  for a9, b9 being Object of B st a9 = a & b9 = b
  for f being Morphism of a,b, f9 being Morphism of b9,a9 st f9 = f
  holds f is retraction iff f9 is coretraction
proof
  let A, B be category such that
A1: A, B are_opposite;
  let a, b be Object of A such that
A2: <^a,b^> <> {} and
A3: <^b,a^> <> {};
  let a9, b9 be Object of B such that
A4: a9 = a and
A5: b9 = b;
A6: <^b9,a9^> = <^a,b^> by A1,A4,A5,Th9;
  <^a9,b9^> = <^b,a^> by A1,A4,A5,Th9;
  hence thesis by A1,A2,A3,A4,A5,A6,Lm1;
end;
