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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	            Relations between UA and Adjunctions      		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export CoUniversalArrow.
Require Export Adjunction.
Set Implicit Arguments.
Unset Strict Implicit.
(* extraction d'un UA et d'un coUA a` partir d'un adjonction *)
Section adj_to_ua.
Variables (C D : Category) (F : Functor D C) (G : Functor C D) (ad : Adj F G).
Variable d : D.
(* Prouver que pour tout d, <F(d),phi(Id(F(d)))> est UA de d vers G *)
Definition Unit_ob := F d.
Definition Unit_arrow := ApAphi ad (Id (F d)).
Definition Unit_arrow_diese (c : C) (f : d --> G c) := ApAphi_inv ad f.
(* qq c:C. qq f:d->Gc. f=phi<d,Fd>(Id(Fd)) o G(phi-1<d,c>(f)) (@) *)
(* remarquons que phi<d,c>(Id(Fd) o phi-1<d,c>(f)) = f *)
(* donc (@) est une instance de Adj_eq3 *)

(* qq c1:C. qq f:d->Gc1. qq f':Fd->c1. *)
(* phi<d,Fd>(Id(Fd)) o G(f') = f (@) *)
(* => f'=phi-1<d,c1>(f) *)
(* appliquer phi-1<d,c1> aux 2 membres de (@) *)
(* En appliquant Adj_eq5, on obtient le re'sultat souhaite' *)
Lemma Unit_UAlaw2 : UA_law2 Unit_arrow Unit_arrow_diese.
Canonical Structure Unit' := Build_IsUA Unit_UAlaw1 Unit_UAlaw2.