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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	             HomFoncteur  Hom(a,-) is Complete        		     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
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Require Export HomFunctor.
Require Export Pres_Limits.
Set Implicit Arguments.
Unset Strict Implicit.
(* exemple : les Hom-Foncteurs preservent toutes les Limites *)
Section funset_pres.
Variables (J C : Category) (F : Functor J C) (l : Limit F) (c : C).
Definition FS_lim := c --> Lim l.
Definition FS_lcone := Limiting_cone l o_C FunSET c.
Section fs_diese.
(* soit X un setoid, soit tau : cone(X,FoH) (i.e. Delta(X) -> FoH) pour tout x:|X|, nous construisons un Tx : cone(c,F) (i.e. Delta(c) -> F) telque Tx(i) = tau(i)(x) *)
Variables (X : SET) (tau : Cone X (F o_F FunSET c)).
Section fs_diese_mor_def.
Variable x : X.
Definition FS_cone_tau (i : J) := tau i x.
(* revient a` prouver: tau(j)(x) = tau(i)(x) o Fg *)
(* s'obtient apre`s simplification de Eq_cone sur tau *)

(* Specifications: *)
Lemma FS_cone_tau_cone_law : Cone_law FS_cone_tau.
Definition FS_cone := Build_Cone FS_cone_tau_cone_law.