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

theorem Th23:
  for A, B being transitive with_units non empty AltCatStr for F
  being contravariant Functor of A, B for o1, o2 being Object of A, a being
  Morphism of o1, o2 st <^o1,o2^> <> {} & <^o2,o1^> <> {} & a is coretraction
  holds F.a is retraction
proof
  let A, B be transitive with_units non empty AltCatStr, F be contravariant
  Functor of A, B, o1, o2 be Object of A, a be Morphism of o1, o2 such that
A1: <^o1,o2^> <> {} & <^o2,o1^> <> {};
  assume a is coretraction;
  then consider b being Morphism of o2, o1 such that
A2: a is_right_inverse_of b;
  take F.b;
  b * a = idm o1 by A2;
  hence (F.a) * (F.b) = F.idm o1 by A1,FUNCTOR0:def 24
    .= idm F.o1 by Th13;
end;
