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(* 02110-1301 USA                                                     *)
(*****************************************************************************)
(*          Projet Coq - Calculus of Inductive Constructions V5.10           *)
(*****************************************************************************)
(*                                                                           *)
(*          Category Theory : Hom (Bi-)Functors (used in Adjunctions)        *)
(*                                                                           *)
(*          Amokrane Saibi May 1994                                          *)
(*                                                                           *)
(*****************************************************************************)
Require Export SET.
Require Export Dual.
Require Export PROD.
Require Export Functor.
Set Implicit Arguments.
Unset Strict Implicit.
(* C(-,G-) et D(F-,-) : D^o x C -> SET *)
(* C(F-,-) *)
Section FunSet2_r.
Variable C D : Category.
(* pour alle'ger les e'critures *)
Section abrev.
Definition OB_l (dxc : POb (Dual D) C) : D := Ob_l dxc.
Variables (d1xc1 d2xc2 : POb (Dual D) C) (foxg : Pmor d1xc1 d2xc2).
Definition HOM_l : OB_l d2xc2 --> OB_l d1xc1 := Hom_l foxg.
Definition Build_POb1 (d : D) (c : C) := Build_POb (A:=Dual D) (B:=C) d c.
Definition Build_Pmor1 (c c' : C) (d d' : D) (f : d' --> d)
(g : c --> c') := Build_Pmor (u:=Build_POb1 d c) (t:=Build_POb1 d' c') f g.
End abrev.
(* *)
Variable F : Functor D C.
Definition FunSET2_r_ob (dxc : POb (Dual D) C) := F (OB_l dxc) --> Ob_r dxc.
Section funset2_r_map_def.
Variable d1xc1 d2xc2 : POb (Dual D) C.
Section funset2_r_mor_def.
Variable foxg : Pmor d1xc1 d2xc2.
Definition FunSET2_r_mor1 (h : FunSET2_r_ob d1xc1) :=
(FMor F (HOM_l foxg) o h) o Hom_r foxg.

Lemma FunSET2_r_map_law1 : Map_law FunSET2_r_mor1.
Canonical Structure FunSET2_r_mor :
Map (FunSET2_r_ob d1xc1) (FunSET2_r_ob d2xc2) := FunSET2_r_map_law1.