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+{-# OPTIONS --without-K --safe #-}
+
+module Functor.Instance.Nat.Pull where
+
+open import Categories.Category.Instance.Nat using (Natop)
+open import Categories.Category.Instance.Setoids using (Setoids)
+open import Categories.Functor using (Functor)
+open import Data.Fin.Base using (Fin)
+open import Data.Nat.Base using (ℕ)
+open import Function.Base using (id; _∘_)
+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 Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
+open import Data.Circuit.Value using (Value)
+open import Data.System.Values Value using (Values)
+
+open Functor
+open Func
+
+_≈_ : {X Y : Setoid 0ℓ 0ℓ} → Rel (X ⟶ₛ Y) 0ℓ
+_≈_ {X} {Y} = Setoid._≈_ (setoid X Y)
+infixr 4 _≈_
+
+private
+  variable A B C : ℕ
+
+
+-- action on objects is Values n (Vector Value 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)
+
+-- Pull flips composition
+Pull-homomorphism
+    : {A B C : ℕ}
+      (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
+
+-- the Pull functor
+Pull : Functor Natop (Setoids 0ℓ 0ℓ)
+F₀ Pull = Values
+F₁ Pull = Pull₁
+identity Pull = Pull-identity
+homomorphism Pull {f = f} {g} {v} = Pull-homomorphism g f {v}
+F-resp-≈ Pull = Pull-resp-≈