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(* 02110-1301 USA                                                     *)


(*****************************************************************************)
(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
(*****************************************************************************)
(*									     *)
(*	           Definition of Terminal by Adjunctions       		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
(*****************************************************************************)


Require Export FunOne.
Require Export Adj_UA.

Set Implicit Arguments.
Unset Strict Implicit.

(* !C a un RAdj => C a un obj terminal *)

Section terminal1_def.

Variable C : Category.

SubClass Terminal1 := RightAdj (FunOne C).

Variable t : Terminal1.

Let ua' := CoUnit t Obone.

Definition Terminal1_ob := CoUA_ob ua'. 

Definition MorT1 (a : C) := CoUA_diese ua' (a':=a) Id_Obone.

Lemma Prf_isTerminal1 : IsTerminal MorT1.
Proof.
(* Goal: @IsTerminal C (@CoUA_ob C One Obone (FunOne C) ua') MorT1 *)
red in |- *; intros a f.
(* Goal: @Equal (@Hom C a (@CoUA_ob C One Obone (FunOne C) ua')) f (MorT1 a) *)
apply (Prf_isCoUA_law2 ua' (f:=Id_Obone) (g:=f)).
(* Goal: @CoUA_eq C One Obone (FunOne C) (@CoUA_ob C One Obone (FunOne C) ua') (@CoUA_mor C One Obone (FunOne C) ua') a Id_Obone f *)
unfold CoUA_eq in |- *; exact I.
Qed.

Canonical Structure Terminal1_to_Terminal := Build_Terminal Prf_isTerminal1.

End terminal1_def.

Coercion Terminal1_to_Terminal : Terminal1 >-> Terminal.
