diff --git a/move_to_lib.hlean b/move_to_lib.hlean
index 990567a..c7ab7ff 100644
--- a/move_to_lib.hlean
+++ b/move_to_lib.hlean
@@ -710,205 +710,6 @@ namespace misc
 
 end misc
 
-namespace category
-
-  definition precategory_Group.{u} [instance] [constructor] : precategory.{u+1 u} Group :=
-  begin
-    fapply precategory.mk,
-    { exact λG H, G →g H },
-    { exact _ },
-    { exact λG H K ψ φ, ψ ∘g φ },
-    { exact λG, gid G },
-    { intros, apply homomorphism_eq, esimp },
-    { intros, apply homomorphism_eq, esimp },
-    { intros, apply homomorphism_eq, esimp }
-  end
-
-
-  definition precategory_AbGroup.{u} [instance] [constructor] : precategory.{u+1 u} AbGroup :=
-  begin
-    fapply precategory.mk,
-    { exact λG H, G →g H },
-    { exact _ },
-    { exact λG H K ψ φ, ψ ∘g φ },
-    { exact λG, gid G },
-    { intros, apply homomorphism_eq, esimp },
-    { intros, apply homomorphism_eq, esimp },
-    { intros, apply homomorphism_eq, esimp }
-  end
-  open iso
-  definition Group_is_iso_of_is_equiv {G H : Group} (φ : G →g H) (H : is_equiv (group_fun φ)) :
-    is_iso φ :=
-  begin
-    fconstructor,
-    { exact (isomorphism.mk φ H)⁻¹ᵍ },
-    { apply homomorphism_eq, rexact left_inv φ },
-    { apply homomorphism_eq, rexact right_inv φ }
-  end
-
-  definition Group_is_equiv_of_is_iso {G H : Group} (φ : G ⟶ H) (Hφ : is_iso φ) :
-    is_equiv (group_fun φ) :=
-  begin
-    fapply adjointify,
-    { exact group_fun φ⁻¹ʰ },
-    { note p := right_inverse φ, exact ap010 group_fun p },
-    { note p := left_inverse φ,  exact ap010 group_fun p }
-  end
-
-  definition Group_iso_equiv (G H : Group) : (G ≅ H) ≃ (G ≃g H) :=
-  begin
-    fapply equiv.MK,
-    { intro φ, induction φ with φ φi, constructor, exact Group_is_equiv_of_is_iso φ _ },
-    { intro v, induction v with φ φe, constructor, exact Group_is_iso_of_is_equiv φ _ },
-    { intro v, induction v with φ φe, apply isomorphism_eq, reflexivity },
-    { intro φ, induction φ with φ φi, apply iso_eq, reflexivity }
-  end
-
-  definition Group_props.{u} {A : Type.{u}} (v : (A → A → A) × (A → A) × A) : Prop.{u} :=
-  begin
-    induction v with m v, induction v with i o,
-    fapply trunctype.mk,
-    { exact is_set A × (Πa, m a o = a) × (Πa, m o a = a) × (Πa b c, m (m a b) c = m a (m b c)) ×
-      (Πa, m (i a) a = o) },
-    { apply is_trunc_of_imp_is_trunc, intro v, induction v with H v,
-      have is_prop (Πa, m a o = a), from _,
-      have is_prop (Πa, m o a = a), from _,
-      have is_prop (Πa b c, m (m a b) c = m a (m b c)), from _,
-      have is_prop (Πa, m (i a) a = o), from _,
-      apply is_trunc_prod }
-  end
-
-  definition Group.sigma_char2.{u} : Group.{u} ≃
-    Σ(A : Type.{u}) (v : (A → A → A) × (A → A) × A), Group_props v :=
-  begin
-    fapply equiv.MK,
-    { intro G, refine ⟨G, _⟩, induction G with G g, induction g with m s ma o om mo i mi,
-      repeat (fconstructor; do 2 try assumption), },
-    { intro v, induction v with x v, induction v with y v, repeat induction y with x y,
-      repeat induction v with x v, constructor, fconstructor, repeat assumption },
-    { intro v, induction v with x v, induction v with y v, repeat induction y with x y,
-      repeat induction v with x v, reflexivity },
-    { intro v, repeat induction v with x v, reflexivity },
-  end
-  open is_trunc
-
-  section
-  local attribute group.to_has_mul group.to_has_inv [coercion]
-
-  theorem inv_eq_of_mul_eq {A : Type} (G H : group A) (p : @mul A G ~2 @mul A H) :
-    @inv A G ~ @inv A H :=
-  begin
-    have foo : Π(g : A), @inv A G g = (@inv A G g * g) * @inv A H g,
-      from λg, !mul_inv_cancel_right⁻¹,
-    cases G with Gs Gm Gh1 G1 Gh2 Gh3 Gi Gh4,
-    cases H with Hs Hm Hh1 H1 Hh2 Hh3 Hi Hh4,
-    change Gi ~ Hi, intro g, have p' : Gm ~2 Hm, from p,
-    calc
-      Gi g = Hm (Hm (Gi g) g) (Hi g) : foo
-       ... = Hm (Gm (Gi g) g) (Hi g) : by rewrite p'
-       ... = Hm G1 (Hi g) : by rewrite Gh4
-       ... = Gm G1 (Hi g) : by rewrite p'
-       ... = Hi g : Gh2
-  end
-
-  theorem one_eq_of_mul_eq {A : Type} (G H : group A)
-    (p : @mul A (group.to_has_mul G) ~2 @mul A (group.to_has_mul H)) :
-    @one A (group.to_has_one G) = @one A (group.to_has_one H) :=
-  begin
-    cases G with Gm Gs Gh1 G1 Gh2 Gh3 Gi Gh4,
-    cases H with Hm Hs Hh1 H1 Hh2 Hh3 Hi Hh4,
-    exact (Hh2 G1)⁻¹ ⬝ (p H1 G1)⁻¹ ⬝ Gh3 H1,
-  end
-  end
-
-  open prod.ops
-  definition group_of_Group_props.{u} {A : Type.{u}} {m : A → A → A} {i : A → A} {o : A}
-    (H : Group_props (m, (i, o))) : group A :=
-  ⦃group, mul := m, inv := i, one := o, is_set_carrier := H.1,
-    mul_one := H.2.1, one_mul := H.2.2.1, mul_assoc := H.2.2.2.1, mul_left_inv := H.2.2.2.2⦄
-
-  theorem Group_eq_equiv_lemma2 {A : Type} {m m' : A → A → A} {i i' : A → A} {o o' : A}
-    (H : Group_props (m, (i, o))) (H' : Group_props (m', (i', o'))) :
-    (m, (i, o)) = (m', (i', o')) ≃ (m ~2 m') :=
-  begin
-    have is_set A, from pr1 H,
-    apply equiv_of_is_prop,
-    { intro p, exact apd100 (eq_pr1 p)},
-    { intro p, apply prod_eq (eq_of_homotopy2 p),
-      apply prod_eq: esimp [Group_props] at *; esimp,
-      { apply eq_of_homotopy,
-        exact inv_eq_of_mul_eq (group_of_Group_props H) (group_of_Group_props H') p },
-      { exact one_eq_of_mul_eq (group_of_Group_props H) (group_of_Group_props H') p }}
-  end
-
-  open sigma.ops
-
-  theorem Group_eq_equiv_lemma {G H : Group}
-    (p : (Group.sigma_char2 G).1 = (Group.sigma_char2 H).1) :
-    ((Group.sigma_char2 G).2 =[p] (Group.sigma_char2 H).2) ≃
-    (is_mul_hom (equiv_of_eq (proof p qed : Group.carrier G = Group.carrier H))) :=
-  begin
-    refine !sigma_pathover_equiv_of_is_prop ⬝e _,
-    induction G with G g, induction H with H h,
-    esimp [Group.sigma_char2] at p, induction p,
-    refine !pathover_idp ⬝e _,
-    induction g with s m ma o om mo i mi, induction h with σ μ μa ε εμ με ι μι,
-    exact Group_eq_equiv_lemma2 (Group.sigma_char2 (Group.mk G (group.mk s m ma o om mo i mi))).2.2
-                                (Group.sigma_char2 (Group.mk G (group.mk σ μ μa ε εμ με ι μι))).2.2
-  end
-
-  definition isomorphism.sigma_char (G H : Group) : (G ≃g H) ≃ Σ(e : G ≃ H), is_mul_hom e :=
-  begin
-    fapply equiv.MK,
-    { intro φ, exact ⟨equiv_of_isomorphism φ, to_respect_mul φ⟩ },
-    { intro v, induction v with e p, exact isomorphism_of_equiv e p },
-    { intro v, induction v with e p, induction e, reflexivity },
-    { intro φ, induction φ with φ H, induction φ, reflexivity },
-  end
-
-  definition Group_eq_equiv (G H : Group) : G = H ≃ (G ≃g H) :=
-  begin
-    refine (eq_equiv_fn_eq_of_equiv Group.sigma_char2 G H) ⬝e _,
-    refine !sigma_eq_equiv ⬝e _,
-    refine sigma_equiv_sigma_right Group_eq_equiv_lemma ⬝e _,
-    transitivity (Σ(e : (Group.sigma_char2 G).1 ≃ (Group.sigma_char2 H).1),
-      @is_mul_hom _ _ _ _ (to_fun e)), apply sigma_ua,
-    exact !isomorphism.sigma_char⁻¹ᵉ
-  end
-
-  definition to_fun_Group_eq_equiv {G H : Group} (p : G = H)
-    : Group_eq_equiv G H p ~ isomorphism_of_eq p :=
-  begin
-    induction p, reflexivity
-  end
-
-  definition Group_eq2 {G H : Group} {p q : G = H}
-    (r : isomorphism_of_eq p ~ isomorphism_of_eq q) : p = q :=
-  begin
-    apply eq_of_fn_eq_fn (Group_eq_equiv G H),
-    apply isomorphism_eq,
-    intro g, refine to_fun_Group_eq_equiv p g ⬝ r g ⬝ (to_fun_Group_eq_equiv q g)⁻¹,
-  end
-
-  definition Group_eq_equiv_Group_iso (G₁ G₂ : Group) : G₁ = G₂ ≃ G₁ ≅ G₂ :=
-  Group_eq_equiv G₁ G₂ ⬝e (Group_iso_equiv G₁ G₂)⁻¹ᵉ
-
-  definition category_Group.{u} : category Group.{u} :=
-  category.mk precategory_Group
-  begin
-    intro G H,
-    apply is_equiv_of_equiv_of_homotopy (Group_eq_equiv_Group_iso G H),
-    intro p, induction p, fapply iso_eq, apply homomorphism_eq, reflexivity
-  end
-
-  definition category_AbGroup : category AbGroup :=
-  category.mk precategory_AbGroup sorry
-
-  definition Grp.{u} [constructor] : Category := category.Mk Group.{u} category_Group
-  definition AbGrp [constructor] : Category := category.Mk AbGroup category_AbGroup
-
-end category
-
 namespace sphere
 
   -- definition constant_sphere_map_sphere {n m : ℕ} (H : n < m) (f : S n →* S m) :
diff --git a/univalent_subcategory.hlean b/univalent_subcategory.hlean
index 83ad751..fca280a 100644
--- a/univalent_subcategory.hlean
+++ b/univalent_subcategory.hlean
@@ -4,20 +4,20 @@ open eq pointed sigma is_equiv equiv fiber algebra group is_trunc function prod
 
 namespace category
 
-section univ_functor
+section univ_subcat
   parameters {C : Precategory} {D : Category} (F : functor C D) (p : is_embedding  F) (q : fully_faithful F) 
   variables  {a b : carrier C}
   include p q
 
-  definition ab_eq_equiv_iso : a = b ≃ a ≅ b := 
+  definition eq_equiv_iso_of_fully_faithful : a = b ≃ a ≅ b := 
     equiv.mk !ap !p                          -- a = b ≃ F a = F b
     ⬝e equiv.mk iso_of_eq !iso_of_path_equiv -- F a = F b ≃ F a ≅ F b
     ⬝e equiv.symm !iso_equiv_F_iso_F         -- F a ≅ F b ≃ a ≅ b
 
-  definition ab_equiv_homot_iso_of_eq : @ab_eq_equiv_iso a b ~ iso_of_eq :=
+  definition eq_equiv_iso_of_fully_faithful_homot : @eq_equiv_iso_of_fully_faithful a b ~ iso_of_eq :=
   begin
     intro r, 
-    esimp [ab_eq_equiv_iso], 
+    esimp [eq_equiv_iso_of_fully_faithful], 
     refine _ ⬝ left_inv (iso_equiv_F_iso_F F _ _) _, 
     apply ap (inv (to_fun !iso_equiv_F_iso_F)), 
     apply symm, 
@@ -25,12 +25,12 @@ section univ_functor
     apply respect_refl
   end
 
-  definition univ_domain : is_univalent C := 
+  definition is_univalent_domain_of_fully_faithful_embedding : is_univalent C := 
   begin
     intros, 
-    apply homotopy_closed ab_eq_equiv_iso ab_equiv_homot_iso_of_eq
+    apply homotopy_closed eq_equiv_iso_of_fully_faithful eq_equiv_iso_of_fully_faithful_homot
   end
-end univ_functor
+end univ_subcat
 
   definition precategory_Group.{u} [instance] [constructor] : precategory.{u+1 u} Group :=
   begin
@@ -165,14 +165,14 @@ repeat (fconstructor; assumption), assumption, intro b,
 
 open sigma.ops
 
-definition sigma_sigma_prod {A} {B C : A→Type} : (Σa, B a × C a) ≃ Σ p : (Σa, B a), C p.1 :=
+definition sigma_prod_equiv_sigma_sigma {A} {B C : A→Type} : (Σa, B a × C a) ≃ Σ p : (Σa, B a), C p.1 :=
 sigma_equiv_sigma_right (λa, !sigma.equiv_prod⁻¹ᵉ) ⬝e !sigma_assoc_equiv
 
-definition ab_group_group_comm (A : Type) : ab_group A ≃ Σ (g : group A), ∀ a b : A, a * b = b * a :=
+definition ab_group_equiv_group_comm (A : Type) : ab_group A ≃ Σ (g : group A), ∀ a b : A, a * b = b * a :=
 begin
 refine !ab_group.sigma_char ⬝e _, 
 refine sigma_equiv_sigma_right AbGroup_Group_props ⬝e _, 
-refine sigma_sigma_prod ⬝e _, 
+refine sigma_prod_equiv_sigma_sigma ⬝e _, 
 apply equiv.symm, apply sigma_equiv_sigma !group.sigma_char, intros, 
 induction a, reflexivity
 end
@@ -264,9 +264,7 @@ induction p,
 
   definition to_fun_Group_eq_equiv {G H : Group} (p : G = H)
     : Group_eq_equiv G H p ~ isomorphism_of_eq p :=
-  begin
-    induction p, reflexivity
-  end
+  begin induction p, reflexivity end
 
   definition Group_eq2 {G H : Group} {p q : G = H}
     (r : isomorphism_of_eq p ~ isomorphism_of_eq q) : p = q :=
@@ -299,7 +297,7 @@ apply is_trunc_equiv_closed, apply equiv.symm,
 apply group.sigma_char
 end
 
-definition ab_group_to_group (A : Type) (g : ab_group A) : group A := _
+definition ab_group_to_group (A  : Type) (g : ab_group A) : group A := _
 
 definition group_comm_to_group (A : Type) : (Σ g : group A, ∀ (a b : A), a*b = b*a) → group A := pr1
 
@@ -314,17 +312,17 @@ induction a', fconstructor, intros, apply sigma_eq,
  apply is_prop.elim, intros, apply is_prop.elim, intros, apply is_prop.elim
 end
 
-definition th (A : Type) : 
-  @ab_group_to_group A ~ group_comm_to_group A ∘ ab_group_group_comm A := 
+definition ab_group_to_group_homot (A : Type) : 
+  @ab_group_to_group A ~ group_comm_to_group A ∘ ab_group_equiv_group_comm A := 
 begin intro, induction x, reflexivity end
 
 definition is_embedding_ab_group_to_group (A : Type) : is_embedding (@ab_group_to_group A) :=
 begin
-apply is_embedding_homotopy_closed_rev (th A), apply is_embedding_compose, 
+apply is_embedding_homotopy_closed_rev (ab_group_to_group_homot A), apply is_embedding_compose, 
 exact is_embedding_group_comm_to_group A, apply is_embedding_of_is_equiv
  end
 
-definition sigma_emb {A} {B C : A → Type} {f : Π a, B a → C a}
+definition is_embedding_total_of_is_embedding_fiber {A} {B C : A → Type} {f : Π a, B a → C a}
   : (∀ a, is_embedding (f a)) → is_embedding (total f) :=
 begin
 intro e, fapply is_embedding_of_is_prop_fiber, intro p, induction p with a c, 
@@ -335,22 +333,22 @@ apply is_prop_fiber_of_is_embedding,
 apply is_trunc_equiv_closed -1 (H⁻¹ᵉ),
 end
 
-definition h2 : AbGroup_to_Group ~ Group_sigma⁻¹ ∘ total ab_group_to_group ∘ AbGroup_sigma :=
+definition AbGroup_to_Group_homot : AbGroup_to_Group ~ Group_sigma⁻¹ ∘ total ab_group_to_group ∘ AbGroup_sigma :=
 begin intro g, induction g, reflexivity end
 
 definition is_embedding_AbGroup_to_Group : is_embedding AbGroup_to_Group :=
 begin
-apply is_embedding_homotopy_closed_rev h2,
+apply is_embedding_homotopy_closed_rev AbGroup_to_Group_homot,
 apply is_embedding_compose, 
 apply is_embedding_of_is_equiv, 
 apply is_embedding_compose,
-apply sigma_emb is_embedding_ab_group_to_group, 
+apply is_embedding_total_of_is_embedding_fiber is_embedding_ab_group_to_group, 
 apply is_embedding_of_is_equiv
 end
 
 definition is_univalent_AbGroup : is_univalent precategory_AbGroup := 
 begin
-apply univ_domain AbGroup_to_Group is_embedding_AbGroup_to_Group, intros, apply is_equiv_id
+apply is_univalent_domain_of_fully_faithful_embedding AbGroup_to_Group is_embedding_AbGroup_to_Group, intros, apply is_equiv_id
 end
 
 definition category_AbGroup : category AbGroup := category.mk precategory_AbGroup is_univalent_AbGroup