3 files changed, 374 insertions, 168 deletions
diff --git a/Functor/Instance/Nat/Pull.agda b/Functor/Instance/Nat/Pull.agda
index b1764d9..c2a06c6 100644
--- a/Functor/Instance/Nat/Pull.agda
+++ b/Functor/Instance/Nat/Pull.agda
@@ -2,80 +2,99 @@
 
 module Functor.Instance.Nat.Pull where
 
+open import Level using (0ℓ)
+
+open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using (Setoids-×)
+
 open import Categories.Category.Instance.Nat using (Natop)
-open import Categories.Category.Instance.Setoids using (Setoids)
 open import Categories.Functor using (Functor)
+open import Category.Construction.CMonoids Setoids-×.symmetric using (CMonoids)
+open import Data.CMonoid using (fromCMonoid)
+open import Data.Circuit.Value using (Monoid)
 open import Data.Fin.Base using (Fin)
+open import Data.Monoid using (fromMonoid)
 open import Data.Nat.Base using (ℕ)
+open import Data.Product using (_,_)
+open import Data.Setoid using (∣_∣; _⇒ₛ_)
+open import Data.System.Values Monoid using (Values; _⊕_; module Object)
+open import Data.Unit using (⊤; tt)
 open import Function.Base using (id; _∘_)
-open import Function.Bundles using (Func; _⟶ₛ_)
+open import Function.Bundles using (Func; _⟶ₛ_; _⟨$⟩_)
 open import Function.Construct.Identity using () renaming (function to Id)
 open import Function.Construct.Setoid using (setoid; _∙_)
-open import Level using (0ℓ)
+open import Object.Monoid.Commutative Setoids-×.symmetric using (CommutativeMonoid; CommutativeMonoid⇒)
 open import Relation.Binary using (Rel; Setoid)
 open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
-open import Data.Circuit.Value using (Monoid)
-open import Data.System.Values Monoid using (Values)
-open import Data.Unit using (⊤; tt)
 
+open CommutativeMonoid using (Carrier; monoid)
+open CommutativeMonoid⇒ using (arr)
 open Functor
 open Func
-
--- Pull takes a natural number n to the setoid Values n
+open Object using (Valuesₘ)
 
 private
 
   variable A B C : ℕ
 
-  _≈_ : {X Y : Setoid 0ℓ 0ℓ} → Rel (X ⟶ₛ Y) 0ℓ
-  _≈_ {X} {Y} = Setoid._≈_ (setoid X Y)
-
-  infixr 4 _≈_
-
-  opaque
+-- Pull takes a natural number n to the commutative monoid Valuesₘ n
 
-    unfolding Values
+Pull₀ : ℕ → CommutativeMonoid
+Pull₀ n = Valuesₘ n
 
-    -- action of Pull on morphisms (contravariant)
-    Pull₁ : (Fin A → Fin B) → Values B ⟶ₛ Values A
-    to (Pull₁ f) i = i ∘ f
-    cong (Pull₁ f) x≗y = x≗y ∘ f
-
-    -- Pull respects identity
-    Pull-identity : Pull₁ id ≈ Id (Values A)
-    Pull-identity {A} = Setoid.refl (Values A)
-
-  opaque
-
-    unfolding Pull₁
-
-    -- Pull flips composition
-    Pull-homomorphism
-        : (f : Fin A → Fin B)
-          (g : Fin B → Fin C)
-        → Pull₁ (g ∘ f) ≈ Pull₁ f ∙ Pull₁ g
-    Pull-homomorphism {A} _ _ = Setoid.refl (Values A)
-
-    -- Pull respects equality
-    Pull-resp-≈
-        : {f g : Fin A → Fin B}
-        → f ≗ g
-        → Pull₁ f ≈ Pull₁ g
-    Pull-resp-≈ f≗g {v} = ≡.cong v ∘ f≗g
+-- action of Pull on morphisms (contravariant)
 
+-- setoid morphism
 opaque
+  unfolding Valuesₘ Values
+  Pullₛ : (Fin A → Fin B) → Carrier (Pull₀ B) ⟶ₛ Carrier (Pull₀ A)
+  Pullₛ f .to x = x ∘ f
+  Pullₛ f .cong x≗y = x≗y ∘ f
 
-  unfolding Pull₁
-
-  Pull-defs : ⊤
-  Pull-defs = tt
+-- monoid morphism
+opaque
+  unfolding _⊕_ Pullₛ
+  Pullₘ : (Fin A → Fin B) → CommutativeMonoid⇒ (Pull₀ B) (Pull₀ A)
+  Pullₘ {A} f = record
+      { monoid⇒ = record
+          { arr = Pullₛ f
+          ; preserves-μ = Setoid.refl (Values A)
+          ; preserves-η = Setoid.refl (Values A)
+          }
+      }
+
+-- Pull respects identity
+opaque
+  unfolding Pullₘ
+  Pull-identity
+      : (open Setoid (Carrier (Pull₀ A) ⇒ₛ Carrier (Pull₀ A)))
+      → arr (Pullₘ id) ≈ Id (Carrier (Pull₀ A))
+  Pull-identity {A} = Setoid.refl (Values A)
 
--- the Pull functor
-Pull : Functor Natop (Setoids 0ℓ 0ℓ)
-F₀ Pull = Values
-F₁ Pull = Pull₁
-identity Pull = Pull-identity
-homomorphism Pull {f = f} {g} = Pull-homomorphism g f
-F-resp-≈ Pull = Pull-resp-≈
+-- Pull flips composition
+opaque
+  unfolding Pullₘ
+  Pull-homomorphism
+      : (f : Fin A → Fin B)
+        (g : Fin B → Fin C)
+        (open Setoid (Carrier (Pull₀ C) ⇒ₛ Carrier (Pull₀ A)))
+      → arr (Pullₘ (g ∘ f)) ≈ arr (Pullₘ f) ∙ arr (Pullₘ g)
+  Pull-homomorphism {A} _ _ = Setoid.refl (Values A)
+
+-- Pull respects equality
+opaque
+  unfolding Pullₘ
+  Pull-resp-≈
+      : {f g : Fin A → Fin B}
+      → f ≗ g
+      → (open Setoid (Carrier (Pull₀ B) ⇒ₛ Carrier (Pull₀ A)))
+      → arr (Pullₘ f) ≈ arr (Pullₘ g)
+  Pull-resp-≈ f≗g {v} = ≡.cong v ∘ f≗g
+
+Pull : Functor Natop CMonoids
+Pull .F₀ = Pull₀
+Pull .F₁ = Pullₘ
+Pull .identity = Pull-identity
+Pull .homomorphism = Pull-homomorphism _ _
+Pull .F-resp-≈ = Pull-resp-≈
 
 module Pull = Functor Pull
diff --git a/Functor/Instance/Nat/Push.agda b/Functor/Instance/Nat/Push.agda
index 8126006..71b9a63 100644
--- a/Functor/Instance/Nat/Push.agda
+++ b/Functor/Instance/Nat/Push.agda
@@ -2,78 +2,115 @@
 
 module Functor.Instance.Nat.Push where
 
-open import Categories.Functor using (Functor)
+open import Level using (0ℓ)
+open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using (Setoids-×)
+
 open import Categories.Category.Instance.Nat using (Nat)
 open import Categories.Category.Instance.Setoids using (Setoids)
-open import Data.Circuit.Merge using (merge; merge-cong₁; merge-cong₂; merge-⁅⁆; merge-preimage)
-open import Data.Fin.Base using (Fin)
+open import Categories.Functor using (Functor)
+open import Category.Construction.CMonoids Setoids-×.symmetric using (CMonoids)
+open import Data.Circuit.Value using (Monoid)
+open import Data.Fin using (Fin)
 open import Data.Fin.Preimage using (preimage; preimage-cong₁)
-open import Data.Nat.Base using (ℕ)
+open import Data.Nat using (ℕ)
+open import Data.Setoid using (∣_∣; _⇒ₛ_)
 open import Data.Subset.Functional using (⁅_⁆)
-open import Function.Base using (id; _∘_)
-open import Function.Bundles using (Func; _⟶ₛ_)
+open import Data.System.Values Monoid using (Values; module Object; _⊕_; <ε>; _≋_; ≋-isEquiv; merge; push; merge-⊕; merge-<ε>; merge-cong; merge-⁅⁆; merge-push; merge-cong₂)
+open import Function using (Func; _⟶ₛ_; _⟨$⟩_; _∘_; id)
 open import Function.Construct.Identity using () renaming (function to Id)
-open import Function.Construct.Setoid using (setoid; _∙_)
-open import Level using (0ℓ)
-open import Relation.Binary using (Rel; Setoid)
+open import Function.Construct.Setoid using (_∙_)
+open import Object.Monoid.Commutative Setoids-×.symmetric using (CommutativeMonoid; CommutativeMonoid⇒)
+open import Relation.Binary using (Setoid)
 open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
-open import Data.Circuit.Value using (Monoid)
-open import Data.System.Values Monoid using (Values)
-open import Data.Unit using (⊤; tt)
 
 open Func
 open Functor
-
--- Push sends a natural number n to Values n
+open Object using (Valuesₘ)
 
 private
 
   variable A B C : ℕ
 
-  _≈_ : {X Y : Setoid 0ℓ 0ℓ} → Rel (X ⟶ₛ Y) 0ℓ
-  _≈_ {X} {Y} = Setoid._≈_ (setoid X Y)
-  infixr 4 _≈_
+  -- Push sends a natural number n to Values n
+  Push₀ : ℕ → CommutativeMonoid
+  Push₀ n = Valuesₘ n
+
+  -- action of Push on morphisms (covariant)
 
   opaque
 
     unfolding Values
 
-    -- action of Push on morphisms (covariant)
-    Push₁ : (Fin A → Fin B) → Values A ⟶ₛ Values B
-    to (Push₁ f) v = merge v ∘ preimage f ∘ ⁅_⁆
-    cong (Push₁ f) x≗y = merge-cong₁ x≗y ∘ preimage f ∘ ⁅_⁆
+    open CommutativeMonoid using (Carrier)
+    open CommutativeMonoid⇒ using (arr)
+
+    -- setoid morphism
+    Pushₛ : (Fin A → Fin B) → Values A ⟶ₛ Values B
+    Pushₛ f .to v = merge v ∘ preimage f ∘ ⁅_⁆
+    Pushₛ f .cong x≗y i = merge-cong (preimage f ⁅ i ⁆) x≗y
+
+  opaque
+
+    unfolding Pushₛ _⊕_
+
+    Push-⊕
+        : (f : Fin A → Fin B)
+        → (xs ys : ∣ Values A ∣)
+        → Pushₛ f ⟨$⟩ (xs ⊕ ys)
+        ≋ (Pushₛ f ⟨$⟩ xs) ⊕ (Pushₛ f ⟨$⟩ ys)
+    Push-⊕ f xs ys i = merge-⊕ xs ys (preimage f ⁅ i ⁆)
+
+    Push-<ε>
+        : (f : Fin A → Fin B)
+        → Pushₛ f ⟨$⟩ <ε> ≋ <ε>
+    Push-<ε> f i = merge-<ε> (preimage f ⁅ i ⁆)
+
+  opaque
+
+    unfolding Push-⊕ ≋-isEquiv Valuesₘ
+
+    -- monoid morphism
+    Pushₘ : (Fin A → Fin B) → CommutativeMonoid⇒ (Valuesₘ A) (Valuesₘ B)
+    Pushₘ f = record
+        { monoid⇒ = record
+            { arr = Pushₛ f
+            ; preserves-μ = Push-⊕ f _ _
+            ; preserves-η = Push-<ε> f
+            }
+        }
 
     -- Push respects identity
-    Push-identity : Push₁ id ≈ Id (Values A)
-    Push-identity {_} {v} = merge-⁅⁆ v
+    Push-identity
+      : (open Setoid (Carrier (Push₀ A) ⇒ₛ Carrier (Push₀ A)))
+      → arr (Pushₘ id) ≈ Id (Carrier (Push₀ A))
+    Push-identity {_} {v} i = merge-⁅⁆ v i
+
+  opaque
+
+    unfolding Pushₘ push
 
     -- Push respects composition
     Push-homomorphism
         : {f : Fin A → Fin B}
           {g : Fin B → Fin C}
-        → Push₁ (g ∘ f) ≈ Push₁ g ∙ Push₁ f
-    Push-homomorphism {f = f} {g} {v} = merge-preimage f v ∘ preimage g ∘ ⁅_⁆
+        → (open Setoid (Carrier (Push₀ A) ⇒ₛ Carrier (Push₀ C)))
+        → arr (Pushₘ (g ∘ f)) ≈ arr (Pushₘ g) ∙ arr (Pushₘ f)
+    Push-homomorphism {f = f} {g} {v} = merge-push f g v
 
     -- Push respects equality
     Push-resp-≈
         : {f g : Fin A → Fin B}
         → f ≗ g
-        → Push₁ f ≈ Push₁ g
+        → (open Setoid (Carrier (Push₀ A) ⇒ₛ Carrier (Push₀ B)))
+        → arr (Pushₘ f) ≈ arr (Pushₘ g)
     Push-resp-≈ f≗g {v} = merge-cong₂ v ∘ preimage-cong₁ f≗g ∘ ⁅_⁆
 
-opaque
-
-  unfolding Push₁
-
-  Push-defs : ⊤
-  Push-defs = tt
-
 -- the Push functor
-Push : Functor Nat (Setoids 0ℓ 0ℓ)
-F₀ Push = Values
-F₁ Push = Push₁
-identity Push = Push-identity
-homomorphism Push = Push-homomorphism
-F-resp-≈ Push = Push-resp-≈
+Push : Functor Nat CMonoids
+Push .F₀ = Push₀
+Push .F₁ = Pushₘ
+Push .identity = Push-identity
+Push .homomorphism = Push-homomorphism
+Push .F-resp-≈ = Push-resp-≈
 
 module Push = Functor Push
diff --git a/Functor/Instance/Nat/System.agda b/Functor/Instance/Nat/System.agda
index 05e1e7b..4a651f2 100644
--- a/Functor/Instance/Nat/System.agda
+++ b/Functor/Instance/Nat/System.agda
@@ -1,110 +1,260 @@
 {-# OPTIONS --without-K --safe #-}
+{-# OPTIONS --hidden-argument-puns #-}
 
 module Functor.Instance.Nat.System where
 
-
 open import Level using (suc; 0ℓ)
 
 open import Categories.Category.Instance.Nat using (Nat)
-open import Categories.Category.Instance.Setoids using (Setoids)
-open import Categories.Functor.Core using (Functor)
+open import Categories.Category using (Category)
+open import Categories.Category.Instance.Cats using (Cats)
+open import Categories.NaturalTransformation.NaturalIsomorphism using (_≃_; niHelper)
+open import Categories.Functor using (Functor; _∘F_) renaming (id to idF)
 open import Data.Circuit.Value using (Monoid)
-open import Data.Fin.Base using (Fin)
-open import Data.Nat.Base using (ℕ)
+open import Data.Fin using (Fin)
+open import Data.Nat using (ℕ)
 open import Data.Product.Base using (_,_; _×_)
-open import Data.System {suc 0ℓ} using (System; _≤_; Systemₛ)
-open import Data.System.Values Monoid using (module ≋)
-open import Data.Unit using (⊤; tt)
-open import Function.Base using (id; _∘_)
-open import Function.Bundles using (Func; _⟶ₛ_; _⟨$⟩_)
+open import Data.Setoid using (∣_∣)
+open import Data.System {suc 0ℓ} using (System; _≤_; Systemₛ; Systems; ≤-refl; ≤-trans; _≈_)
+open import Data.System.Values Monoid using (module ≋; module Object; Values; ≋-isEquiv)
+open import Relation.Binary using (Setoid)
+open import Function using (Func; _⟶ₛ_; _⟨$⟩_; _∘_; id)
 open import Function.Construct.Identity using () renaming (function to Id)
 open import Function.Construct.Setoid using (_∙_)
 open import Functor.Instance.Nat.Pull using (Pull)
 open import Functor.Instance.Nat.Push using (Push)
 open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
 
+open import Category.Instance.Setoids.SymmetricMonoidal {0ℓ} {0ℓ} using (Setoids-×)
+open import Category.Construction.CMonoids Setoids-×.symmetric using (CMonoids)
+open import Object.Monoid.Commutative Setoids-×.symmetric using (CommutativeMonoid; CommutativeMonoid⇒)
+
+open CommutativeMonoid⇒ using (arr)
+open Object using (Valuesₘ)
 open Func
 open Functor
 open _≤_
 
 private
-
   variable A B C : ℕ
 
-  opaque
+opaque
+  unfolding Valuesₘ ≋-isEquiv
+  map : (Fin A → Fin B) → System A → System B
+  map {A} {B} f X = let open System X in record
+      { S = S
+      ; fₛ = fₛ ∙ arr (Pull.₁ f)
+      ; fₒ = arr (Push.₁ f) ∙ fₒ
+      }
+
+opaque
+  unfolding map
+  open System
+  map-≤ : (f : Fin A → Fin B) {X Y : System A} → X ≤ Y → map f X ≤ map f Y
+  ⇒S (map-≤ f x≤y) = ⇒S x≤y
+  ≗-fₛ (map-≤ f x≤y) = ≗-fₛ x≤y ∘ to (arr (Pull.₁ f))
+  ≗-fₒ (map-≤ f x≤y) = cong (arr (Push.₁ f)) ∘ ≗-fₒ x≤y
+
+opaque
+  unfolding map-≤
+  map-≤-refl
+      : (f : Fin A → Fin B)
+      → {X : System A}
+      → map-≤ f (≤-refl {A} {X}) ≈ ≤-refl
+  map-≤-refl f {X} = Setoid.refl (S (map f X))
+
+opaque
+  unfolding map-≤
+  map-≤-trans
+      : (f : Fin A → Fin B)
+      → {X Y Z : System A}
+      → {h : X ≤ Y}
+      → {g : Y ≤ Z}
+      → map-≤ f (≤-trans h g) ≈ ≤-trans (map-≤ f h) (map-≤ f g)
+  map-≤-trans f {Z = Z} = Setoid.refl (S (map f Z))
 
-    map : (Fin A → Fin B) → System A → System B
-    map f X = let open System X in record
-        { S = S
-        ; fₛ = fₛ ∙ Pull.₁ f
-        ; fₒ = Push.₁ f ∙ fₒ
+opaque
+  unfolding map-≤
+  map-≈
+      : (f : Fin A → Fin B)
+      → {X Y : System A}
+      → {g h : X ≤ Y}
+      → h ≈ g
+      → map-≤ f h ≈ map-≤ f g
+  map-≈ f h≈g = h≈g
+
+Sys₁ : (Fin A → Fin B) → Functor (Systems A) (Systems B)
+Sys₁ {A} {B} f = record
+    { F₀ = map f
+    ; F₁ = λ C≤D → map-≤ f C≤D
+    ; identity = map-≤-refl f
+    ; homomorphism = map-≤-trans f
+    ; F-resp-≈ = map-≈ f
+    }
+
+opaque
+  unfolding map
+  map-id-≤ : (X : System A) → map id X ≤ X
+  map-id-≤ X .⇒S = Id (S X)
+  map-id-≤ X .≗-fₛ i s = cong (fₛ X) Pull.identity
+  map-id-≤ X .≗-fₒ s = Push.identity
+
+opaque
+  unfolding map
+  map-id-≥ : (X : System A) → X ≤ map id X
+  map-id-≥ X .⇒S = Id (S X)
+  map-id-≥ X .≗-fₛ i s = cong (fₛ X) (≋.sym Pull.identity)
+  map-id-≥ X .≗-fₒ s = ≋.sym Push.identity
+
+opaque
+  unfolding map-≤ map-id-≤
+  map-id-comm
+      : {X Y : System A}
+        (f : X ≤ Y)
+      → ≤-trans (map-≤ id f) (map-id-≤ Y) ≈ ≤-trans (map-id-≤ X) f
+  map-id-comm {Y} f = Setoid.refl (S Y)
+
+opaque
+
+  unfolding map-id-≤ map-id-≥
+
+  map-id-isoˡ
+      : (X : System A)
+      → ≤-trans (map-id-≤ X) (map-id-≥ X) ≈ ≤-refl
+  map-id-isoˡ X = Setoid.refl (S X)
+
+  map-id-isoʳ
+      : (X : System A)
+      → ≤-trans (map-id-≥ X) (map-id-≤ X) ≈ ≤-refl
+  map-id-isoʳ X = Setoid.refl (S X)
+
+Sys-identity : Sys₁ {A} id ≃ idF
+Sys-identity = niHelper record
+    { η = map-id-≤
+    ; η⁻¹ = map-id-≥
+    ; commute = map-id-comm
+    ; iso = λ X → record
+        { isoˡ = map-id-isoˡ X
+        ; isoʳ = map-id-isoʳ X
+        }
+    }
+
+opaque
+  unfolding map
+  map-∘-≤
+      : (f : Fin A → Fin B)
+        (g : Fin B → Fin C)
+        (X : System A)
+      → map (g ∘ f) X ≤ map g (map f X)
+  map-∘-≤ f g X .⇒S = Id (S X)
+  map-∘-≤ f g X .≗-fₛ i s = cong (fₛ X) Pull.homomorphism
+  map-∘-≤ f g X .≗-fₒ s = Push.homomorphism
+
+opaque
+  unfolding map
+  map-∘-≥
+      : (f : Fin A → Fin B)
+        (g : Fin B → Fin C)
+        (X : System A)
+      → map g (map f X) ≤ map (g ∘ f) X
+  map-∘-≥ f g X .⇒S = Id (S X)
+  map-∘-≥ f g X .≗-fₛ i s = cong (fₛ X) (≋.sym Pull.homomorphism)
+  map-∘-≥ f g X .≗-fₒ s = ≋.sym Push.homomorphism
+
+Sys-homo
+    : (f : Fin A → Fin B)
+      (g : Fin B → Fin C)
+    → Sys₁ (g ∘ f) ≃ Sys₁ g ∘F Sys₁ f
+Sys-homo {A} f g = niHelper record
+    { η = map-∘-≤ f g
+    ; η⁻¹ = map-∘-≥ f g
+    ; commute = map-∘-comm f g
+    ; iso = λ X → record
+        { isoˡ = isoˡ X
+        ; isoʳ = isoʳ X
         }
+    }
+  where
+    opaque
+      unfolding map-∘-≤ map-≤
+      map-∘-comm
+          : (f : Fin A → Fin B)
+            (g : Fin B → Fin C)
+          → {X Y : System A}
+            (X≤Y : X ≤ Y)
+          → ≤-trans (map-≤ (g ∘ f) X≤Y) (map-∘-≤ f g Y)
+          ≈ ≤-trans (map-∘-≤ f g X) (map-≤ g (map-≤ f X≤Y))
+      map-∘-comm f g {Y} X≤Y = Setoid.refl (S Y)
+    module _ (X : System A) where
+      opaque
+        unfolding map-∘-≤ map-∘-≥
+        isoˡ : ≤-trans (map-∘-≤ f g X) (map-∘-≥ f g X) ≈ ≤-refl
+        isoˡ = Setoid.refl (S X)
+        isoʳ : ≤-trans (map-∘-≥ f g X) (map-∘-≤ f g X) ≈ ≤-refl
+        isoʳ = Setoid.refl (S X)
 
-    ≤-cong : (f : Fin A → Fin B) {X Y : System A} → Y ≤ X → map f Y ≤ map f X
-    ⇒S (≤-cong f x≤y) = ⇒S x≤y
-    ≗-fₛ (≤-cong f x≤y) = ≗-fₛ x≤y ∘ to (Pull.₁ f)
-    ≗-fₒ (≤-cong f x≤y) = cong (Push.₁ f) ∘ ≗-fₒ x≤y
 
-    System₁ : (Fin A → Fin B) → Systemₛ A ⟶ₛ Systemₛ B
-    to (System₁ f) = map f
-    cong (System₁ f) (x≤y , y≤x) = ≤-cong f x≤y , ≤-cong f y≤x
+module _ {f g : Fin A → Fin B} (f≗g : f ≗ g) (X : System A) where
 
   opaque
 
-    unfolding System₁
-
-    id-x≤x : {X : System A} → System₁ id ⟨$⟩ X ≤ X
-    ⇒S (id-x≤x) = Id _
-    ≗-fₛ (id-x≤x {_} {x}) i s = cong (System.fₛ x) Pull.identity
-    ≗-fₒ (id-x≤x {A} {x}) s = Push.identity
-
-    x≤id-x : {x : System A} → x ≤ System₁ id ⟨$⟩ x
-    ⇒S x≤id-x = Id _
-    ≗-fₛ (x≤id-x {A} {x}) i s = cong (System.fₛ x) (≋.sym Pull.identity)
-    ≗-fₒ (x≤id-x {A} {x}) s = ≋.sym Push.identity
-
-    System-homomorphism
-        : {f : Fin A → Fin B}
-          {g : Fin B → Fin C} 
-          {X : System A}
-        → System₁ (g ∘ f) ⟨$⟩ X ≤ System₁ g ⟨$⟩ (System₁ f ⟨$⟩ X)
-        × System₁ g ⟨$⟩ (System₁ f ⟨$⟩ X) ≤ System₁ (g ∘ f) ⟨$⟩ X
-    System-homomorphism {f = f} {g} {X} = left , right
-      where
-        open System X
-        left : map (g ∘ f) X ≤ map g (map f X)
-        left .⇒S = Id S
-        left .≗-fₛ i s = cong fₛ Pull.homomorphism
-        left .≗-fₒ s = Push.homomorphism
-        right : map g (map f X) ≤ map (g ∘ f) X
-        right .⇒S = Id S
-        right .≗-fₛ i s = cong fₛ (≋.sym Pull.homomorphism)
-        right .≗-fₒ s = ≋.sym Push.homomorphism
-
-    System-resp-≈
-        : {f g : Fin A → Fin B}
-        → f ≗ g
-        → {X : System A}
-        → System₁ f ⟨$⟩ X ≤ System₁ g ⟨$⟩ X
-        × System₁ g ⟨$⟩ X ≤ System₁ f ⟨$⟩ X
-    System-resp-≈ {A} {B} {f = f} {g} f≗g {X} = both f≗g , both (≡.sym ∘ f≗g)
-      where
-        open System X
-        both : {f g : Fin A → Fin B} → f ≗ g → map f X ≤ map g X
-        both f≗g .⇒S = Id S
-        both f≗g .≗-fₛ i s = cong fₛ (Pull.F-resp-≈ f≗g {i})
-        both {f} {g} f≗g .≗-fₒ s = Push.F-resp-≈ f≗g
+    unfolding map
+
+    map-cong-≤ : map f X ≤ map g X
+    map-cong-≤ .⇒S = Id (S X)
+    map-cong-≤ .≗-fₛ i s = cong (fₛ X) (Pull.F-resp-≈ f≗g)
+    map-cong-≤ .≗-fₒ s = Push.F-resp-≈ f≗g
+
+    map-cong-≥ : map g X ≤ map f X
+    map-cong-≥ .⇒S = Id (S X)
+    map-cong-≥ .≗-fₛ i s = cong (fₛ X) (Pull.F-resp-≈ (≡.sym ∘ f≗g))
+    map-cong-≥ .≗-fₒ s = Push.F-resp-≈ (≡.sym ∘ f≗g)
 
 opaque
-  unfolding System₁
-  Sys-defs : ⊤
-  Sys-defs = tt
-
-Sys : Functor Nat (Setoids (suc 0ℓ) (suc 0ℓ))
-Sys .F₀ = Systemₛ
-Sys .F₁ = System₁
-Sys .identity = id-x≤x , x≤id-x
-Sys .homomorphism {x = X} = System-homomorphism {X = X}
-Sys .F-resp-≈ = System-resp-≈
+  unfolding map-≤ map-cong-≤
+  map-cong-comm
+      : {f g : Fin A → Fin B}
+        (f≗g : f ≗ g)
+        {X Y : System A}
+        (h : X ≤ Y)
+      → ≤-trans (map-≤ f h) (map-cong-≤ f≗g Y)
+      ≈ ≤-trans (map-cong-≤ f≗g X) (map-≤ g h)
+  map-cong-comm f≗g {Y} h = Setoid.refl (S Y)
+
+opaque
+
+  unfolding map-cong-≤
+
+  map-cong-isoˡ
+      : {f g : Fin A → Fin B}
+        (f≗g : f ≗ g)
+        (X : System A)
+      → ≤-trans (map-cong-≤ f≗g X) (map-cong-≥ f≗g X) ≈ ≤-refl
+  map-cong-isoˡ f≗g X = Setoid.refl (S X)
+
+  map-cong-isoʳ
+      : {f g : Fin A → Fin B}
+        (f≗g : f ≗ g)
+        (X : System A)
+      → ≤-trans (map-cong-≥ f≗g X) (map-cong-≤ f≗g X) ≈ ≤-refl
+  map-cong-isoʳ f≗g X = Setoid.refl (S X)
+
+Sys-resp-≈ : {f g : Fin A → Fin B} → f ≗ g → Sys₁ f ≃ Sys₁ g
+Sys-resp-≈ f≗g = niHelper record
+    { η = map-cong-≤ f≗g
+    ; η⁻¹ = map-cong-≥ f≗g
+    ; commute = map-cong-comm f≗g
+    ; iso = λ X → record
+        { isoˡ = map-cong-isoˡ f≗g X
+        ; isoʳ = map-cong-isoʳ f≗g X
+        }
+    }
+
+Sys : Functor Nat (Cats (suc 0ℓ) (suc 0ℓ) 0ℓ)
+Sys .F₀ = Systems
+Sys .F₁ = Sys₁
+Sys .identity = Sys-identity
+Sys .homomorphism = Sys-homo _ _
+Sys .F-resp-≈ = Sys-resp-≈
 
 module Sys = Functor Sys