2 files changed, 97 insertions, 5 deletions
diff --git a/Category/Cocomplete/Finitely/Bundle.agda b/Category/Cocomplete/Finitely/Bundle.agda
index af40895..74f434f 100644
--- a/Category/Cocomplete/Finitely/Bundle.agda
+++ b/Category/Cocomplete/Finitely/Bundle.agda
@@ -1,12 +1,12 @@
 {-# OPTIONS --without-K --safe #-}
 module Category.Cocomplete.Finitely.Bundle where
 
-open import Level
+open import Categories.Category.Cocartesian.Bundle using (CocartesianCategory)
 open import Categories.Category.Core using (Category)
 open import Categories.Category.Cocomplete.Finitely using (FinitelyCocomplete)
 open import Category.Cocomplete.Finitely.SymmetricMonoidal using (module FinitelyCocompleteSymmetricMonoidal)
 open import Categories.Category.Monoidal.Bundle using (SymmetricMonoidalCategory)
-
+open import Level using (_⊔_; suc)
 
 record FinitelyCocompleteCategory o ℓ e : Set (suc (o ⊔ ℓ ⊔ e)) where
 
@@ -17,11 +17,20 @@ record FinitelyCocompleteCategory o ℓ e : Set (suc (o ⊔ ℓ ⊔ e)) where
   open Category U public
   open FinitelyCocomplete finCo public
 
-  open FinitelyCocompleteSymmetricMonoidal finCo using (+-monoidal; +-symmetric)
+  open FinitelyCocompleteSymmetricMonoidal finCo
+    using ()
+    renaming (+-monoidal to monoidal; +-symmetric to symmetric)
+    public
 
   symmetricMonoidalCategory : SymmetricMonoidalCategory o ℓ e
   symmetricMonoidalCategory = record
       { U = U
-      ; monoidal = +-monoidal
-      ; symmetric = +-symmetric
+      ; monoidal = monoidal
+      ; symmetric = symmetric
+      }
+
+  cocartesianCategory : CocartesianCategory o ℓ e
+  cocartesianCategory = record
+      { U = U
+      ; cocartesian = cocartesian
       }
diff --git a/Category/Cocomplete/Finitely/Product.agda b/Category/Cocomplete/Finitely/Product.agda
new file mode 100644
index 0000000..25dc346
--- /dev/null
+++ b/Category/Cocomplete/Finitely/Product.agda
@@ -0,0 +1,83 @@
+{-# OPTIONS --without-K --safe #-}
+
+open import Categories.Category.Core using (Category)
+open import Level using (Level)
+
+module Category.Cocomplete.Finitely.Product {o ℓ e : Level} {𝒞 𝒟 : Category o ℓ e} where
+
+open import Categories.Category using (_[_,_])
+open import Categories.Category.Cocomplete.Finitely using (FinitelyCocomplete)
+open import Categories.Category.Cocartesian using (Cocartesian; BinaryCoproducts)
+open import Categories.Category.Product using (Product)
+open import Categories.Diagram.Coequalizer using (Coequalizer)
+open import Categories.Object.Coproduct using (Coproduct)
+open import Categories.Object.Initial using (IsInitial; Initial)
+open import Data.Product.Base using (_,_; _×_; dmap; zip; map)
+
+Initial-× : Initial 𝒞 → Initial 𝒟 → Initial (Product 𝒞 𝒟)
+Initial-× initial-𝒞 initial-𝒟 = record
+    { ⊥ = 𝒞.⊥ , 𝒟.⊥
+    ; ⊥-is-initial = record
+        { ! = 𝒞.! , 𝒟.!
+        ; !-unique = dmap 𝒞.!-unique 𝒟.!-unique
+        }
+    }
+  where
+    module 𝒞 = Initial initial-𝒞
+    module 𝒟 = Initial initial-𝒟
+
+Coproducts-× : BinaryCoproducts 𝒞 → BinaryCoproducts 𝒟 → BinaryCoproducts (Product 𝒞 𝒟)
+Coproducts-× coproducts-𝒞 coproducts-𝒟 = record { coproduct = coproduct }
+  where
+    coproduct : ∀ {(A₁ , B₁) (A₂ , B₂) : _ × _} → Coproduct (Product 𝒞 𝒟) (A₁ , B₁) (A₂ , B₂)
+    coproduct = record
+        { A+B = 𝒞.A+B , 𝒟.A+B
+        ; i₁ = 𝒞.i₁ , 𝒟.i₁
+        ; i₂ = 𝒞.i₂ , 𝒟.i₂
+        ; [_,_] = zip 𝒞.[_,_] 𝒟.[_,_]
+        ; inject₁ = 𝒞.inject₁ , 𝒟.inject₁
+        ; inject₂ = 𝒞.inject₂ , 𝒟.inject₂
+        ; unique = zip 𝒞.unique 𝒟.unique
+        }
+      where
+        module Coprod {𝒞} (coprods : BinaryCoproducts 𝒞) where
+          open BinaryCoproducts coprods using (coproduct)
+          open coproduct public
+        module 𝒞 = Coprod coproducts-𝒞
+        module 𝒟 = Coprod coproducts-𝒟
+
+Coequalizer-×
+    : (∀ {A} {B} (f g : 𝒞 [ A , B ]) → Coequalizer 𝒞 f g)
+    → (∀ {A} {B} (f g : 𝒟 [ A , B ]) → Coequalizer 𝒟 f g)
+    → ∀ {A} {B} {C} {D} ((f₁ , g₁) (f₂ , g₂) : 𝒞 [ A , B ] × 𝒟 [ C , D ])
+    → Coequalizer (Product 𝒞 𝒟) (f₁ , g₁) (f₂ , g₂)
+Coequalizer-× coequalizer-𝒞 coequalizer-𝒟 (f₁ , g₁) (f₂ , g₂) = record
+    { arr = 𝒞.arr , 𝒟.arr
+    ; isCoequalizer = record
+        { equality = 𝒞.equality , 𝒟.equality
+        ; coequalize = map 𝒞.coequalize 𝒟.coequalize
+        ; universal = 𝒞.universal , 𝒟.universal
+        ; unique = map 𝒞.unique 𝒟.unique
+        }
+    }
+  where
+    module 𝒞 = Coequalizer (coequalizer-𝒞 f₁ f₂)
+    module 𝒟 = Coequalizer (coequalizer-𝒟 g₁ g₂)
+
+Cocartesian-× : Cocartesian 𝒞 → Cocartesian 𝒟 → Cocartesian (Product 𝒞 𝒟)
+Cocartesian-× cocartesian-𝒞 cocartesian-𝒟 = record
+    { initial = Initial-× 𝒞.initial 𝒟.initial
+    ; coproducts = Coproducts-× 𝒞.coproducts 𝒟.coproducts
+    }
+  where
+    module 𝒞 = Cocartesian cocartesian-𝒞
+    module 𝒟 = Cocartesian cocartesian-𝒟
+
+FinitelyCocomplete-× : FinitelyCocomplete 𝒞 → FinitelyCocomplete 𝒟 → FinitelyCocomplete (Product 𝒞 𝒟)
+FinitelyCocomplete-× finitelyCocomplete-𝒞 finitelyCocomplete-𝒟 = record
+    { cocartesian = Cocartesian-× 𝒞.cocartesian 𝒟.cocartesian
+    ; coequalizer = Coequalizer-× 𝒞.coequalizer 𝒟.coequalizer
+    }
+  where
+    module 𝒞 = FinitelyCocomplete finitelyCocomplete-𝒞
+    module 𝒟 = FinitelyCocomplete finitelyCocomplete-𝒟