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(** * Kan extensions assemble into functors *)
Require Import Category.Core Functor.Core.
Require Import KanExtensions.Core.
Require Import Adjoint.UniversalMorphisms.Core.
Require Import FunctorCategory.Core.
Require Import Adjoint.Core.

Set Implicit Arguments.
Generalizable All Variables.

Section kan_extension_functors.
  Context `{Funext}.
  Variables C C' D : PreCategory.
  Variable p : object (C -> C').

  (** ** Left Kan extension functor *)
  Section lan.
    Context `(has_left_kan_extensions
              : forall h : object (C -> D),
                  @IsLeftKanExtensionAlong _ _ _ _ p h (left_kan_extensions h)).

    Definition left_kan_extension_functor
    : Functor (C -> D) (C' -> D)
      := functor__of__initial_morphism has_left_kan_extensions.

    Definition left_kan_extension_adjunction
    : left_kan_extension_functor -| pullback_along D p
      := adjunction__of__initial_morphism has_left_kan_extensions.
  End lan.
  (** ** Right Kan extension functor *)
  Section ran.
    Context `(has_right_kan_extensions
              : forall h : object (C -> D),
                  @IsRightKanExtensionAlong _ _ _ _ p h (right_kan_extensions h)).

    Definition right_kan_extension_functor
    : Functor (C -> D) (C' -> D)
      := functor__of__terminal_morphism has_right_kan_extensions.

    Definition right_kan_extension_adjunction
    : pullback_along D p -| right_kan_extension_functor
      := adjunction__of__terminal_morphism has_right_kan_extensions.
  End ran.
End kan_extension_functors.