From 05cbf6f56bce1d45876630fe29b694dc57942e9c Mon Sep 17 00:00:00 2001
From: Jacques Comeaux <jacquesrcomeaux@protonmail.com>
Date: Thu, 13 Nov 2025 13:24:21 -0600
Subject: Add adjunction between free monoid and forget

---
 NaturalTransformation/Instance/EmptyList.agda | 32 ++++++++++++++++++---------
 1 file changed, 22 insertions(+), 10 deletions(-)

(limited to 'NaturalTransformation/Instance/EmptyList.agda')

diff --git a/NaturalTransformation/Instance/EmptyList.agda b/NaturalTransformation/Instance/EmptyList.agda
index 9a558a2..0e069d2 100644
--- a/NaturalTransformation/Instance/EmptyList.agda
+++ b/NaturalTransformation/Instance/EmptyList.agda
@@ -1,23 +1,35 @@
 {-# OPTIONS --without-K --safe #-}
 
-open import Level using (Level)
+open import Level using (Level; _⊔_)
 
 module NaturalTransformation.Instance.EmptyList {c ℓ : Level} where
 
-import Function.Construct.Constant as Const
-
-open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
+open import Categories.Category.Instance.SingletonSet using () renaming (SingletonSetoid to ⊤ₛ)
 open import Categories.Functor using (Functor)
-open import Categories.Category.Instance.SingletonSet using (SingletonSetoid)
 open import Categories.Functor.Construction.Constant using (const)
-open import Data.List using ([])
+open import Categories.NaturalTransformation using (NaturalTransformation; ntHelper)
+open import Data.Opaque.List using (Listₛ; []ₛ; mapₛ)
+open import Data.Setoid using (_⇒ₛ_)
+open import Function using (_⟶ₛ_)
+open import Function.Construct.Constant using () renaming (function to Const)
+open import Function.Construct.Setoid using (_∙_)
 open import Functor.Instance.List {c} {ℓ} using (List)
 open import Relation.Binary using (Setoid)
 
-module List = Functor List
+opaque
+
+  unfolding []ₛ
+
+  map-[]ₛ : {A B : Setoid c ℓ}
+      → (f : A ⟶ₛ B)
+      → (open Setoid (⊤ₛ ⇒ₛ Listₛ B))
+      → []ₛ ≈ mapₛ f ∙ []ₛ
+  map-[]ₛ {_} {B} f = refl
+    where
+      open Setoid (List.₀ B)
 
-⊤⇒[] : NaturalTransformation (const SingletonSetoid) List
+⊤⇒[] : NaturalTransformation (const ⊤ₛ) List
 ⊤⇒[] = ntHelper record
-    { η = λ X → Const.function SingletonSetoid (List.₀ X) []
-    ; commute = λ {_} {B} f → Setoid.refl (List.₀ B)
+    { η = λ X → []ₛ
+    ; commute = map-[]ₛ
     }
-- 
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