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

theorem
  for D being full non empty subcategory of C for o1, o2 being Object of
C, p1, p2 being Object of D for m being Morphism of o1, o2, n being Morphism of
p1, p2 st p1 = o1 & p2 = o2 & m = n & <^p1,p2^> <> {} & <^p2,p1^> <> {} holds (
  m is retraction implies n is retraction) & (m is coretraction implies n is
  coretraction) & (m is iso implies n is iso)
proof
  let D be full non empty subcategory of C, o1, o2 be Object of C, p1, p2 be
  Object of D, m be Morphism of o1, o2, n be Morphism of p1, p2;
  assume that
A1: p1 = o1 & p2 = o2 and
A2: m = n and
A3: <^p1,p2^> <> {} & <^p2,p1^> <> {};
  thus
A4: m is retraction implies n is retraction
  proof
    assume m is retraction;
    then consider B being Morphism of o2, o1 such that
A5: B is_right_inverse_of m;
    reconsider B1 = B as Morphism of p2, p1 by A1,ALTCAT_2:28;
    take B1;
    thus thesis by A1,A2,A3,A5,Th38;
  end;
  thus
A6: m is coretraction implies n is coretraction
  proof
    assume m is coretraction;
    then consider B being Morphism of o2, o1 such that
A7: B is_left_inverse_of m;
    reconsider B1 = B as Morphism of p2, p1 by A1,ALTCAT_2:28;
    take B1;
    thus thesis by A1,A2,A3,A7,Th38;
  end;
  assume m is iso;
  hence thesis by A3,A4,A6,ALTCAT_3:5,6;
end;
