diff --git a/hott/algebra/homotopy_group.hlean b/hott/algebra/homotopy_group.hlean
index 0eb2cee0a..ab8d25185 100644
--- a/hott/algebra/homotopy_group.hlean
+++ b/hott/algebra/homotopy_group.hlean
@@ -161,22 +161,21 @@ namespace eq
   @(is_equiv_trunc_functor 0 _) (is_equiv_apn n f H)
 
   definition homotopy_group_succ_in_natural (n : ℕ) {A B : Type*} (f : A →* B) :
-    homotopy_group_succ_in n B ∘* π→[n + 1] f ~*
-    π→[n] (Ω→ f) ∘* homotopy_group_succ_in n A :=
+    psquare (homotopy_group_succ_in n A) (homotopy_group_succ_in n B)
+            (π→[n + 1] f) (π→[n] (Ω→ f)) :=
   begin
-    refine !ptrunc_functor_pcompose⁻¹* ⬝* _ ⬝* !ptrunc_functor_pcompose,
-    exact ptrunc_functor_phomotopy 0 (apn_succ_phomotopy_in n f)
+    exact ptrunc_functor_psquare 0 (loopn_succ_in_natural n f),
   end
 
   definition homotopy_group_succ_in_natural_unpointed (n : ℕ) {A B : Type*} (f : A →* B) :
     hsquare (homotopy_group_succ_in n A) (homotopy_group_succ_in n B) (π→[n+1] f) (π→[n] (Ω→ f)) :=
-  (homotopy_group_succ_in_natural n f)⁻¹*
+  homotopy_group_succ_in_natural n f
 
   definition is_equiv_homotopy_group_functor_ap1 (n : ℕ) {A B : Type*} (f : A →* B)
     [is_equiv (π→[n + 1] f)] : is_equiv (π→[n] (Ω→ f)) :=
   have is_equiv (π→[n] (Ω→ f) ∘ homotopy_group_succ_in n A),
   from is_equiv_of_equiv_of_homotopy (equiv.mk (π→[n+1] f) _ ⬝e homotopy_group_succ_in n B)
-                                     (homotopy_group_succ_in_natural n f),
+                                     (homotopy_group_succ_in_natural n f)⁻¹*,
   is_equiv.cancel_right (homotopy_group_succ_in n A) _
 
   definition tinverse [constructor] {X : Type*} : π[1] X →* π[1] X :=
diff --git a/hott/homotopy/LES_of_homotopy_groups.hlean b/hott/homotopy/LES_of_homotopy_groups.hlean
index 9efa2a01c..8068627b9 100644
--- a/hott/homotopy/LES_of_homotopy_groups.hlean
+++ b/hott/homotopy/LES_of_homotopy_groups.hlean
@@ -272,7 +272,7 @@ namespace chain_complex
 
   definition pid_or_pinverse_add4_rev (n : ℕ) :
     pid_or_pinverse (n + 4) ~* !pinverse ∘* Ω→(pid_or_pinverse (n + 1)) :=
-  !ap1_pcompose_pinverse
+  !pinverse_natural
 
   theorem fiber_sequence_phomotopy_loop_spaces : Π(n : ℕ),
     fiber_sequence_pequiv_loop_spaces n ∘* fiber_sequence_fun n ~*
@@ -300,7 +300,7 @@ namespace chain_complex
       xrewrite [loop_spaces_fun_add3, pid_or_pinverse_add4, to_pmap_pequiv_trans],
       refine _ ⬝* !passoc⁻¹*,
       refine _ ⬝* pwhisker_left _ !passoc⁻¹*,
-      refine _ ⬝* pwhisker_left _ (pwhisker_left _ !ap1_pcompose_pinverse),
+      refine _ ⬝* pwhisker_left _ (pwhisker_left _ !pinverse_natural),
       refine !passoc⁻¹* ⬝* _ ⬝* !passoc ⬝* !passoc,
       apply pwhisker_right,
       refine !ap1_pcompose⁻¹* ⬝* _ ⬝* !ap1_pcompose ⬝* pwhisker_right _ !ap1_pcompose,
@@ -345,7 +345,7 @@ namespace chain_complex
       apply pwhisker_left,
       refine !ap1_pcompose ⬝* _ ⬝* !passoc ⬝* !passoc,
       apply pwhisker_right,
-      refine _ ⬝* pwhisker_right _ !ap1_pcompose_pinverse,
+      refine _ ⬝* pwhisker_right _ !pinverse_natural,
       refine _ ⬝* !passoc⁻¹*,
       refine !pcompose_pid⁻¹* ⬝* pwhisker_left _ _,
       symmetry, apply pinverse_pinverse
@@ -364,7 +364,7 @@ namespace chain_complex
       replace (k + 4) with (k + 1 + 3),
       rewrite [loop_spaces_fun_add3],
       refine !passoc⁻¹* ⬝* _ ⬝* !passoc⁻¹*,
-      refine _ ⬝* pwhisker_left _ !ap1_pcompose_pinverse,
+      refine _ ⬝* pwhisker_left _ !pinverse_natural,
       refine _ ⬝* !passoc,
       apply pwhisker_right,
       refine !ap1_pcompose⁻¹* ⬝* _ ⬝* !ap1_pcompose,
diff --git a/hott/homotopy/susp.hlean b/hott/homotopy/susp.hlean
index 24df3f986..7d8eba65e 100644
--- a/hott/homotopy/susp.hlean
+++ b/hott/homotopy/susp.hlean
@@ -213,6 +213,8 @@ namespace susp
     { reflexivity }
   end
 
+  notation `⅀→`:(max+5) := susp_functor
+
   definition is_equiv_susp_functor [constructor] (f : X →* Y) [Hf : is_equiv f]
     : is_equiv (susp_functor f) :=
   susp.is_equiv_functor f
@@ -245,6 +247,8 @@ namespace susp
     { reflexivity },
   end
 
+  notation `⅀⇒`:(max+5) := susp_functor_phomotopy
+
   definition susp_functor_pid (A : Type*) : susp_functor (pid A) ~* pid (susp A) :=
   begin
     fapply phomotopy.mk,
@@ -266,8 +270,9 @@ namespace susp
   end
 
   definition loop_susp_unit_natural (f : X →* Y)
-    : loop_susp_unit Y ∘* f ~* Ω→ (susp_functor f) ∘* loop_susp_unit X :=
+    : psquare (loop_susp_unit X) (loop_susp_unit Y) f (Ω→ (susp_functor f)) :=
   begin
+    apply ptranspose,
     induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
     fapply phomotopy.mk,
     { intro x', symmetry,
@@ -294,7 +299,7 @@ namespace susp
   end
 
   definition loop_susp_counit_natural (f : X →* Y)
-    : f ∘* loop_susp_counit X ~* loop_susp_counit Y ∘* (susp_functor (ap1 f)) :=
+    : psquare (loop_susp_counit X) (loop_susp_counit Y) (⅀→ (Ω→ f)) f :=
   begin
     induction X with X x, induction Y with Y y, induction f with f pf, esimp at *, induction pf,
     fconstructor,
@@ -360,7 +365,7 @@ namespace susp
 
   definition loop_susp_intro_natural {X Y Z : Type*} (g : susp Y →* Z) (f : X →* Y) :
     loop_susp_intro (g ∘* susp_functor f) ~* loop_susp_intro g ∘* f :=
-  pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_susp_unit_natural⁻¹* ⬝*
+  pwhisker_right _ !ap1_pcompose ⬝* !passoc ⬝* pwhisker_left _ !loop_susp_unit_natural ⬝*
   !passoc⁻¹*
 
   definition susp_adjoint_loop_right_inv {X Y : Type*} (g : X →* Ω Y) :
@@ -368,7 +373,7 @@ namespace susp
   begin
     refine !pwhisker_right !ap1_pcompose ⬝* _,
     refine !passoc ⬝* _,
-    refine !pwhisker_left !loop_susp_unit_natural⁻¹* ⬝* _,
+    refine !pwhisker_left !loop_susp_unit_natural ⬝* _,
     refine !passoc⁻¹* ⬝* _,
     refine !pwhisker_right !loop_susp_counit_unit ⬝* _,
     apply pid_pcompose
@@ -403,7 +408,8 @@ namespace susp
     { apply eq_pathover, refine !elim_merid ⬝ph _ ⬝hp !ap_constant⁻¹, exact square_of_eq !con.right_inv⁻¹ }
   end
 
-  definition susp_functor_pconst [constructor] (X Y : Type*) : susp_functor (pconst X Y) ~* pconst (susp X) (susp Y) :=
+  definition susp_functor_pconst [constructor] (X Y : Type*) :
+    susp_functor (pconst X Y) ~* pconst (susp X) (susp Y) :=
   begin
     fapply phomotopy.mk,
     { exact susp_functor_pconst_homotopy },
@@ -422,7 +428,7 @@ namespace susp
   definition loop_susp_pintro_natural_left (f : X' →* X) :
     psquare (loop_susp_pintro X Y) (loop_susp_pintro X' Y)
             (ppcompose_right (susp_functor f)) (ppcompose_right f) :=
-  !pap1_natural_left ⬝h* ppcompose_right_psquare (loop_susp_unit_natural f)⁻¹*
+  !pap1_natural_left ⬝h* ppcompose_right_psquare (loop_susp_unit_natural f)
 
   definition loop_susp_pintro_natural_right (f : Y →* Y') :
     psquare (loop_susp_pintro X Y) (loop_susp_pintro X Y')
diff --git a/hott/types/pointed.hlean b/hott/types/pointed.hlean
index f0b1bab35..74980714e 100644
--- a/hott/types/pointed.hlean
+++ b/hott/types/pointed.hlean
@@ -309,13 +309,6 @@ namespace pointed
     : pinverse X p⁻¹ = (pinverse X p)⁻¹ :=
   idp
 
-  definition ap1_pcompose_pinverse [constructor] {X Y : Type*} (f : X →* Y) :
-    Ω→ f ∘* pinverse X ~* pinverse Y ∘* Ω→ f :=
-  phomotopy.mk (ap1_gen_inv f (respect_pt f) (respect_pt f))
-    abstract begin
-      induction Y with Y y₀, induction f with f f₀, esimp at * ⊢, induction f₀, reflexivity
-    end end
-
   definition is_equiv_pcast [instance] {A B : Type*} (p : A = B) : is_equiv (pcast p) :=
   !is_equiv_cast
 
@@ -580,6 +573,8 @@ namespace pointed
   definition ap1_phomotopy {f g : A →* B} (p : f ~* g) : Ω→ f ~* Ω→ g :=
   pap Ω→ p
 
+  notation `Ω⇒`:(max+5) := ap1_phomotopy
+
   definition ap1_phomotopy_refl {X Y : Type*} (f : X →* Y) :
     ap1_phomotopy (phomotopy.refl f) = phomotopy.refl (Ω→ f) :=
   !pap_refl
@@ -676,11 +671,6 @@ namespace pointed
     (r : p = q) : ptransport B p ~* ptransport B q :=
   phomotopy.mk (λb, ap (λp, transport B p b) r) begin induction r, apply idp_con end
 
-  definition pnatural_square {A B : Type} (X : B → Type*) {f g : A → B}
-    (h : Πa, X (f a) →* X (g a)) {a a' : A} (p : a = a') :
-    h a' ∘* ptransport X (ap f p) ~* ptransport X (ap g p) ∘* h a :=
-  by induction p; exact !pcompose_pid ⬝* !pid_pcompose⁻¹*
-
   definition apn_pid [constructor] {A : Type*} (n : ℕ) : apn n (pid A) ~* pid (Ω[n] A) :=
   begin
     induction n with n IH,
@@ -730,12 +720,6 @@ namespace pointed
     { reflexivity}
   end
 
-  definition pcast_commute [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
-    {a₁ a₂ : A} (p : a₁ = a₂) : pcast (ap C p) ∘* f a₁ ~* f a₂ ∘* pcast (ap B p) :=
-  phomotopy.mk
-    begin induction p, reflexivity end
-    begin induction p, esimp, refine !idp_con ⬝ !idp_con ⬝ !ap_id⁻¹ end
-
   /- pointed equivalences -/
 
   structure pequiv (A B : Type*) :=
@@ -868,11 +852,6 @@ namespace pointed
   definition peap {A B : Type*} (F : Type* → Type*) (p : A ≃* B) : F A ≃* F B :=
   pequiv_of_pmap (pcast (ap F (eq_of_pequiv p))) begin cases eq_of_pequiv p, apply is_equiv_id end
 
-  -- rename pequiv_of_eq_natural
-  definition pequiv_of_eq_commute [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
-    {a₁ a₂ : A} (p : a₁ = a₂) : pequiv_of_eq (ap C p) ∘* f a₁ ~* f a₂ ∘* pequiv_of_eq (ap B p) :=
-  pcast_commute f p
-
   -- definition pequiv.eta_expand [constructor] {A B : Type*} (f : A ≃* B) : A ≃* B :=
   -- pequiv.mk' f (to_pinv f) (pequiv.to_pinv2 f) (pright_inv f) _
 
@@ -1178,27 +1157,6 @@ namespace pointed
       ... = Ω[2+n] A                                 : eq_of_pequiv !loopn_add
       ... = Ω[n+2] A                                 : by rewrite [algebra.add.comm]
 
-  definition apn_succ_phomotopy_in (n : ℕ) (f : A →* B) :
-    loopn_succ_in n B ∘* Ω→[n + 1] f ~* Ω→[n] (Ω→ f) ∘* loopn_succ_in n A :=
-  begin
-    induction n with n IH,
-    { reflexivity},
-    { exact !ap1_pcompose⁻¹* ⬝* ap1_phomotopy IH ⬝* !ap1_pcompose}
-  end
-
-  definition loopn_succ_in_natural {A B : Type*} (n : ℕ) (f : A →* B) :
-    loopn_succ_in n B ∘* Ω→[n+1] f ~* Ω→[n] (Ω→ f) ∘* loopn_succ_in n A :=
-  !apn_succ_phomotopy_in
-
-  definition loopn_succ_in_inv_natural {A B : Type*} (n : ℕ) (f : A →* B) :
-    Ω→[n + 1] f ∘* (loopn_succ_in n A)⁻¹ᵉ* ~* (loopn_succ_in n B)⁻¹ᵉ* ∘* Ω→[n] (Ω→ f):=
-  begin
-    apply pinv_right_phomotopy_of_phomotopy,
-    refine _ ⬝* !passoc⁻¹*,
-    apply phomotopy_pinv_left_of_phomotopy,
-    apply apn_succ_phomotopy_in
-  end
-
   section psquare
   /-
     Squares of pointed maps
@@ -1206,7 +1164,7 @@ namespace pointed
     We treat expressions of the form
       psquare f g h k :≡ k ∘* f ~* g ∘* h
     as squares, where f is the top, g is the bottom, h is the left face and k is the right face.
-    Then the following are operations on squares
+    These squares are very useful for naturality squares
   -/
 
   variables {A' A₀₀ A₂₀ A₄₀ A₀₂ A₂₂ A₄₂ A₀₄ A₂₄ A₄₄ : Type*}
@@ -1226,6 +1184,9 @@ namespace pointed
   definition phomotopy_of_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : f₂₁ ∘* f₁₀ ~* f₁₂ ∘* f₀₁ :=
   p
 
+  definition hsquare_of_psquare (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : hsquare f₁₀ f₁₂ f₀₁ f₂₁ :=
+  to_homotopy p
+
   definition phdeg_square {f f' : A →* A'} (p : f ~* f') : psquare !pid !pid f f' :=
   !pcompose_pid ⬝* p⁻¹* ⬝* !pid_pcompose⁻¹*
   definition pvdeg_square {f f' : A →* A'} (p : f ~* f') : psquare f f' !pid !pid :=
@@ -1233,9 +1194,9 @@ namespace pointed
 
 
   variables (f₁₀ f₁₂ f₀₁ f₂₁)
-  definition hpconst_square : psquare !pconst !pconst f₀₁ f₂₁ :=
+  definition phconst_square : psquare !pconst !pconst f₀₁ f₂₁ :=
   !pcompose_pconst ⬝* !pconst_pcompose⁻¹*
-  definition vpconst_square : psquare f₁₀ f₁₂ !pconst !pconst :=
+  definition pvconst_square : psquare f₁₀ f₁₂ !pconst !pconst :=
   !pconst_pcompose ⬝* !pcompose_pconst⁻¹*
   definition phrefl : psquare !pid !pid f₀₁ f₀₁ := phdeg_square phomotopy.rfl
   definition pvrefl : psquare f₁₀ f₁₀ !pid !pid := pvdeg_square phomotopy.rfl
@@ -1243,6 +1204,9 @@ namespace pointed
   definition phrfl : psquare !pid !pid f₀₁ f₀₁ := phrefl f₀₁
   definition pvrfl : psquare f₁₀ f₁₀ !pid !pid := pvrefl f₁₀
 
+  definition ptranspose (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) : psquare f₀₁ f₂₁ f₁₀ f₁₂ :=
+  p⁻¹*
+
   definition phconcat (p : psquare f₁₀ f₁₂ f₀₁ f₂₁) (q : psquare f₃₀ f₃₂ f₂₁ f₄₁) :
     psquare (f₃₀ ∘* f₁₀) (f₃₂ ∘* f₁₂) f₀₁ f₄₁ :=
   !passoc⁻¹* ⬝* pwhisker_right f₁₀ q ⬝* !passoc ⬝* pwhisker_left f₃₂ p ⬝* !passoc⁻¹*
@@ -1300,4 +1264,40 @@ namespace pointed
 
   end psquare
 
+  definition pinverse_natural [constructor] {X Y : Type*} (f : X →* Y) :
+    psquare (pinverse X) (pinverse Y) (Ω→ f) (Ω→ f) :=
+  phomotopy.mk (ap1_gen_inv f (respect_pt f) (respect_pt f))
+    abstract begin
+      induction Y with Y y₀, induction f with f f₀, esimp at * ⊢, induction f₀, reflexivity
+    end end
+
+  definition pcast_natural [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
+    {a₁ a₂ : A} (p : a₁ = a₂) : psquare (pcast (ap B p)) (pcast (ap C p)) (f a₁) (f a₂) :=
+  phomotopy.mk
+    begin induction p, reflexivity end
+    begin induction p, exact whisker_left idp !ap_id end
+
+  definition pequiv_of_eq_natural [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
+    {a₁ a₂ : A} (p : a₁ = a₂) :
+    psquare (pequiv_of_eq (ap B p)) (pequiv_of_eq (ap C p)) (f a₁) (f a₂) :=
+  pcast_natural f p
+
+  definition loopn_succ_in_natural {A B : Type*} (n : ℕ) (f : A →* B) :
+    psquare (loopn_succ_in n A) (loopn_succ_in n B) (Ω→[n+1] f) (Ω→[n] (Ω→ f)) :=
+  begin
+    induction n with n IH,
+    { exact phomotopy.rfl },
+    { exact ap1_psquare IH }
+  end
+
+  definition loopn_succ_in_inv_natural {A B : Type*} (n : ℕ) (f : A →* B) :
+    psquare (loopn_succ_in n A)⁻¹ᵉ* (loopn_succ_in n B)⁻¹ᵉ* (Ω→[n] (Ω→ f)) (Ω→[n + 1] f) :=
+  (loopn_succ_in_natural n f)⁻¹ʰ*
+
+  definition pnatural_square {A B : Type} (X : B → Type*) {f g : A → B}
+    (h : Πa, X (f a) →* X (g a)) {a a' : A} (p : a = a') :
+   psquare (ptransport X (ap f p)) (ptransport X (ap g p)) (h a) (h a') :=
+  by induction p; exact !pcompose_pid ⬝* !pid_pcompose⁻¹*
+
+
 end pointed