(** * Universal objects *)
Require Import Category.Core Category.Morphisms.
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Set Universe Polymorphism.
Set Implicit Arguments.
Generalizable All Variables.
Set Asymmetric Patterns.

Local Open Scope category_scope.
Local Open Scope morphism_scope.

(** ** Definition of "unique up to unique isomorphism" *)

Definition unique_up_to_unique_isomorphism (C : PreCategory) (P : C -> Type) :=
  forall x (_ : P x) x' (_ : P x'),
    { c : Contr (morphism C x x')
    | IsIsomorphism (center (morphism C x x')) }.

(** ** Terminal objects *)
(** A terminal object is an object with a unique morphism from every
    other object. *)
Notation IsTerminalObject C x :=
  (forall x' : object C, Contr (morphism C x' x)).

Record TerminalObject (C : PreCategory) :=
  {
    object_terminal :> C;
    isterminal_object_terminal :> IsTerminalObject C object_terminal
  }.

Existing Instance isterminal_object_terminal.

(** ** Initial objects *)
(** An initial object is an object with a unique morphism from every
    other object. *)
Notation IsInitialObject C x :=
  (forall x' : object C, Contr (morphism C x x')).

Record InitialObject (C : PreCategory) :=
  {
    object_initial :> C;
    isinitial_object_initial :> IsInitialObject C object_initial
  }.

Existing Instance isinitial_object_initial.

Arguments unique_up_to_unique_isomorphism [C] P.

(** ** Initial and terminal objects are unique up to unique isomorphism *)
Section CategoryObjectsTheorems.
  Variable C : PreCategory.

  Local Ltac unique :=
    repeat first [ intro
                 | exists _
                 | exists (center (morphism C _ _))
                 | etransitivity; [ symmetry | ]; apply contr ].

  (** The terminal object is unique up to unique isomorphism. *)
  Theorem terminal_object_unique
  : unique_up_to_unique_isomorphism (fun x => IsTerminalObject C x).
C: PreCategory
unique_up_to_unique_isomorphism
  (fun x : C => IsTerminalObject C x)
  Proof.
C: PreCategory
unique_up_to_unique_isomorphism
  (fun x : C => IsTerminalObject C x)
    unique.
  Qed.

  (** The initial object is unique up to unique isomorphism. *)
  Theorem initial_object_unique
  : unique_up_to_unique_isomorphism (fun x => IsInitialObject C x).
C: PreCategory
unique_up_to_unique_isomorphism
  (fun x : C => IsInitialObject C x)
  Proof.
C: PreCategory
unique_up_to_unique_isomorphism
  (fun x : C => IsInitialObject C x)
    unique.
  Qed.
End CategoryObjectsTheorems.