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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	                    Yoneda's Lemma                     		     *)
(*									     *)
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(*									     *)
(*                           A.SAIBI   May 95				     *)
(*									     *)
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Require Export HomFunctor_NT.
Require Export CatFunct.
Require Export Functor_dup1.
Set Implicit Arguments.
Unset Strict Implicit.
Section yoneda_lemma.
Variable C : Category.
(* nous voulons de'finir un iso entre NatFun et Ev *)
(* Nous devons d'abord construire une NT YLphi : NatFun -> Ev *)
(* YLphi(a,F) : NaFtun(C(a,-),F) -> F(a) dans SET i.e. que c'est une Map *)
Section ylphi_tau_def.
Variables (a : C) (F : Functor C SET).
Definition YLphi_arrow (T : NT (FunSET a) F) := T a (Id a).

(* Specifications: *)
Lemma YLphi_arrow_map_law : Map_law''0 YLphi_arrow.
Canonical Structure YLphi_tau := Build_Map''0 YLphi_arrow_map_law.