reserve x,y for set;

theorem
  for A being category, B being non empty subcategory of A holds B,B
  are_isomorphic_under id A
proof
  let A be category, B be non empty subcategory of A;
  set F = id A;
  thus B is subcategory of A & B is subcategory of A;
  take G = id B;
  thus G is bijective;
  hereby
    let a be Object of B, a1 be Object of A;
    assume a = a1;
    hence G.a = a1 by FUNCTOR0:29
      .= F.a1 by FUNCTOR0:29;
  end;
  let b,c be Object of B, b1,c1 be Object of A such that
A1: <^b,c^> <> {} and
A2: b = b1 & c = c1;
  let f be Morphism of b,c, f1 be Morphism of b1,c1 such that
A3: f = f1;
A4: <^b,c^> c= <^b1,c1^> & f in <^b,c^> by A1,A2,ALTCAT_2:31;
A5: F.b1 = b1 & F.c1 = c1 by FUNCTOR0:29;
  thus G.f = f by A1,FUNCTOR0:31
    .= F.f1 by A3,A4,FUNCTOR0:31
    .= Morph-Map(F,b1,c1).f1 by A4,A5,FUNCTOR0:def 15;
end;
