Further reduce dependency on impossible facts.

Provide the data for the product in the category of categories without
requiring such a category to actually exist

src/Cat/Categories/Cat.agda

@@ -62,8 +62,7 @@ module _ (ℓ ℓ' : Level) where
 
 -- The following to some extend depends on the category of categories being a
 -- category. In some places it may not actually be needed, however.
-module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
-  module _ (ℂ 𝔻 : Category ℓ ℓ') where
+module CatProducts {ℓ ℓ' : Level} (ℂ 𝔻 : Category ℓ ℓ') where
   private
     :Object: = Object ℂ × Object 𝔻
     :Arrow:  : :Object: → :Object: → Set ℓ'
@@ -84,44 +83,45 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
     RawCategory._∘_ :rawProduct: = _:⊕:_
     open RawCategory :rawProduct:
 
-      module C = Category ℂ
-      module D = Category 𝔻
+    module ℂ = Category ℂ
+    module 𝔻 = Category 𝔻
     open import Cubical.Sigma
-      issSet : {A B : RawCategory.Object :rawProduct:} → isSet (Arrow A B)
-      issSet = setSig {sA = C.arrowsAreSets} {sB = λ x → D.arrowsAreSets}
-      ident' : IsIdentity :𝟙:
-      ident'
-        = Σ≡ (fst C.isIdentity) (fst D.isIdentity)
-        , Σ≡ (snd C.isIdentity) (snd D.isIdentity)
-      postulate univalent : Univalence.Univalent :rawProduct: ident'
+    arrowsAreSets : ArrowsAreSets -- {A B : RawCategory.Object :rawProduct:} → isSet (Arrow A B)
+    arrowsAreSets = setSig {sA = ℂ.arrowsAreSets} {sB = λ x → 𝔻.arrowsAreSets}
+    isIdentity : IsIdentity :𝟙:
+    isIdentity
+      = Σ≡ (fst ℂ.isIdentity) (fst 𝔻.isIdentity)
+      , Σ≡ (snd ℂ.isIdentity) (snd 𝔻.isIdentity)
+    postulate univalent : Univalence.Univalent :rawProduct: isIdentity
     instance
       :isCategory: : IsCategory :rawProduct:
-        IsCategory.isAssociative :isCategory: = Σ≡ C.isAssociative D.isAssociative
-        IsCategory.isIdentity :isCategory: = ident'
-        IsCategory.arrowsAreSets :isCategory: = issSet
+      IsCategory.isAssociative :isCategory: = Σ≡ ℂ.isAssociative 𝔻.isAssociative
+      IsCategory.isIdentity :isCategory: = isIdentity
+      IsCategory.arrowsAreSets :isCategory: = arrowsAreSets
       IsCategory.univalent :isCategory: = univalent
 
-      :product: : Category ℓ ℓ'
-      Category.raw :product: = :rawProduct:
+  obj : Category ℓ ℓ'
+  Category.raw obj = :rawProduct:
 
-      proj₁ : Functor :product: ℂ
+  proj₁ : Functor obj ℂ
   proj₁ = record
     { raw = record { func* = fst ; func→ = fst }
     ; isFunctor = record { isIdentity = refl ; isDistributive = refl }
     }
 
-      proj₂ : Functor :product: 𝔻
+  proj₂ : Functor obj 𝔻
   proj₂ = record
     { raw = record { func* = snd ; func→ = snd }
     ; isFunctor = record { isIdentity = refl ; isDistributive = refl }
     }
 
   module _ {X : Category ℓ ℓ'} (x₁ : Functor X ℂ) (x₂ : Functor X 𝔻) where
-        x : Functor X :product:
+    private
+      x : Functor X obj
       x = record
         { raw = record
-            { func* = λ x → x₁ .func* x , x₂ .func* x
-            ; func→ = λ x → func→ x₁ x , func→ x₂ x
+          { func* = λ x → x₁.func* x , x₂.func* x
+          ; func→ = λ x → x₁.func→ x , x₂.func→ x
           }
         ; isFunctor = record
           { isIdentity   = Σ≡ x₁.isIdentity x₂.isIdentity
@@ -147,27 +147,30 @@ module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
       isUniq : F[ proj₁ ∘ x ] ≡ x₁ × F[ proj₂ ∘ x ] ≡ x₂
       isUniq = isUniqL , isUniqR
 
-        uniq : ∃![ x ] (F[ proj₁ ∘ x ] ≡ x₁ × F[ proj₂ ∘ x ] ≡ x₂)
-        uniq = x , isUniq
+    isProduct : ∃![ x ] (F[ proj₁ ∘ x ] ≡ x₁ × F[ proj₂ ∘ x ] ≡ x₂)
+    isProduct = x , isUniq
 
+module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
+  private
     Catℓ = Cat ℓ ℓ' unprovable
+  module _ (ℂ 𝔻 : Category ℓ ℓ') where
+    private
+      module P = CatProducts ℂ 𝔻
 
       instance
-      isProduct : IsProduct Catℓ proj₁ proj₂
-      isProduct = uniq
+        isProduct : IsProduct Catℓ P.proj₁ P.proj₂
+        isProduct = P.isProduct
 
     product : Product {ℂ = Catℓ} ℂ 𝔻
     product = record
-      { obj = :product:
-      ; proj₁ = proj₁
-      ; proj₂ = proj₂
+      { obj = P.obj
+      ; proj₁ = P.proj₁
+      ; proj₂ = P.proj₂
       }
 
-module _ {ℓ ℓ' : Level} (unprovable : IsCategory (RawCat ℓ ℓ')) where
-  Catt = Cat ℓ ℓ' unprovable
   instance
-    hasProducts : HasProducts Catt
-    hasProducts = record { product = product unprovable }
+    hasProducts : HasProducts Catℓ
+    hasProducts = record { product = product }
 
 -- Basically proves that `Cat ℓ ℓ` is cartesian closed.
 module _ (ℓ : Level) (unprovable : IsCategory (RawCat ℓ ℓ)) where