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defn: Opposite of product category is product of opposites #575

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defn: Opposite of product category is product of opposites
jajaperson Dec 29, 2025
c3e4f55
fixup: Move ×^op into separate file, use iso instead
jajaperson Dec 30, 2025
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1 change: 0 additions & 1 deletion src/Cat/Instances/Product.lagda.md
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Expand Up @@ -135,7 +135,6 @@ _nt×_ α β .is-natural (c , d) (c' , d') (f , g) = Σ-pathp
```
-->


## Univalence

Isomorphisms in functor categories admit a short description, too: They
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46 changes: 46 additions & 0 deletions src/Cat/Instances/Product/Opposite.lagda.md
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<!--
```agda
open import Cat.Functor.Equivalence
open import Cat.Functor.Properties
open import Cat.Instances.Product
open import Cat.Functor.Base
open import Cat.Prelude

import Cat.Reasoning
```
-->

```agda
module Cat.Instances.Product.Opposite {o₁ h₁ o₂ h₂ : Level}
{C : Precategory o₁ h₁} {D : Precategory o₁ h₁}
where
```

<!--
```agda
open Precategory
open Functor
```
-->

# Opposite product category {defines="opposite-product-category"}

As one might expect, taking the [[opposite category]] of a [[product category]]
agrees with the product of opposite categories. Rather than showing
equality we construct an [[isomorphism of precategories]].

```agda
×^op : Functor ((C ×ᶜ D)^op) (C ^op ×ᶜ D ^op)
×^op .F₀ x = x
×^op .F₁ f = f
×^op .F-id = refl
×^op .F-∘ f g = refl

×^op-is-iso : is-precat-iso ×^op
×^op-is-iso = iso has-is-ff has-is-iso where
has-is-ff : Cat.Functor.Properties.is-fully-faithful ×^op
has-is-ff = id-equiv

has-is-iso : is-equiv (F₀ ×^op)
has-is-iso = id-equiv
```