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(*                                                                    *)
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(* 02110-1301 USA                                                     *)
(*****************************************************************************)
(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
(*****************************************************************************)
(*		                                 			     *)
(*	     The Special Adjoint Functor Theorem (Freyd's Theorem)	     *)
(*                      ( The necessary condition )                          *)
(*									     *)
(*****************************************************************************)
(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
(*****************************************************************************)
Require Export Limit.
Require Export Adj_UA.
Require Export Single.
Require Export FAFT_SSC2.
Set Implicit Arguments.
Unset Strict Implicit.
(****************************************)
(* CONDITION NECESSAIRE                 *)
(****************************************)
Section freyd_th_1.
Variables (A X : Category) (G : Functor A X) (A_c : Complete A)
(la : LeftAdj G).
Variable x : X.
Let I := UnitType.
Let a (i : I) := match i with
| Elt => Adjoint la x
end.
Let f (i : I) :=
match i as i' return (Carrier (x --> G (a i'))) with
| Elt => UA_mor (Unit la x)
end.
Section cd2.
Variables (r : A) (h : x --> G r).
Let i := Elt.
Let t : a i --> r := ApAphi_inv la h.

Lemma FT_cd2_law : h =_S f i o FMor G t.
End freyd_th_1.