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------------------------------------------------------------------------
-- The Agda standard library
--
-- Functors
------------------------------------------------------------------------

-- Note that currently the functor laws are not included here.

{-# OPTIONS --cubical-compatible --safe #-}

module Category.Functor where

open import Function hiding (Morphism)
open import Level

open import Relation.Binary.PropositionalEquality

private
  variable
     ℓ′ ℓ″ : Level
    A B X Y : Set 

record RawFunctor (F : Set   Set ℓ′) : Set (suc   ℓ′) where
  infixl 4 _<$>_ _<$_
  infixl 1 _<&>_

  field
    _<$>_ : (A  B)  F A  F B

  _<$_ : A  F B  F A
  x <$ y = const x <$> y

  _<&>_ : F A  (A  B)  F B
  _<&>_ = flip _<$>_

-- A functor morphism from F₁ to F₂ is an operation op such that
-- op (F₁ f x) ≡ F₂ f (op x)

record Morphism {F₁ : Set   Set ℓ′} {F₂ : Set   Set ℓ″}
                (fun₁ : RawFunctor F₁)
                (fun₂ : RawFunctor F₂) : Set (suc   ℓ′  ℓ″) where
  open RawFunctor
  field
    op     : F₁ X  F₂ X
    op-<$> : (f : X  Y) (x : F₁ X) 
             op (fun₁ ._<$>_ f x)  fun₂ ._<$>_ f (op x)