Factor out objects

src/Cat/Categories/Cat.agda

@@ -156,7 +156,7 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
       RawProduct.proj₁ rawProduct = P.proj₁
       RawProduct.proj₂ rawProduct = P.proj₂
 
-      isProduct : IsProduct Catℓ rawProduct
+      isProduct : IsProduct Catℓ _ _ rawProduct
       IsProduct.isProduct isProduct = P.isProduct
 
     product : Product Catℓ ℂ 𝔻

src/Cat/Categories/Sets.agda

@@ -70,7 +70,7 @@ module _ {ℓ : Level} where
         RawProduct.proj₁ rawProduct = Data.Product.proj₁
         RawProduct.proj₂ rawProduct = Data.Product.proj₂
 
-        isProduct : IsProduct 𝓢 rawProduct
+        isProduct : IsProduct 𝓢 _ _ rawProduct
         IsProduct.isProduct isProduct {X = X} f g
           = (f &&& g) , lem {0X = X} f g
 

src/Cat/Category/Product.agda

@@ -10,14 +10,15 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
 
   open Category ℂ
 
-  record RawProduct (A B : Object) : Set (ℓa ⊔ ℓb) where
+  module _ (A B : Object) where
+    record RawProduct : Set (ℓa ⊔ ℓb) where
       no-eta-equality
       field
         obj : Object
         proj₁ : ℂ [ obj , A ]
         proj₂ : ℂ [ obj , B ]
 
-  record IsProduct {A B : Object} (raw : RawProduct A B) : Set (ℓa ⊔ ℓb) where
+    record IsProduct (raw : RawProduct) : Set (ℓa ⊔ ℓb) where
       open RawProduct raw public
       field
         isProduct : ∀ {X : Object} (x₁ : ℂ [ X , A ]) (x₂ : ℂ [ X , B ])
@@ -28,10 +29,10 @@ module _ {ℓa ℓb : Level} (ℂ : Category ℓa ℓb) where
         → ℂ [ X , obj ]
       _P[_×_] π₁ π₂ = P.proj₁ (isProduct π₁ π₂)
 
-  record Product (A B : Object) : Set (ℓa ⊔ ℓb) where
+    record Product : Set (ℓa ⊔ ℓb) where
       field
-      raw        : RawProduct A B
-      isProduct  : IsProduct {A} {B} raw
+        raw        : RawProduct
+        isProduct  : IsProduct raw
 
       open IsProduct isProduct public