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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	                       Adjunctions                		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export NatIso.
Require Export HomFunctor2.
Set Implicit Arguments.
Unset Strict Implicit.
Section adj_def.
Variables (C D : Category) (F : Functor D C) (G : Functor C D).
SubClass Adj := NatIso (FunSET2_r F) (FunSET2_l G).
Variables (phi : NT (FunSET2_r F) (FunSET2_l G))
(phi_1 : NT (FunSET2_l G) (FunSET2_r F)).
Variable phi_iso : AreNatIsos phi phi_1.
Definition Build_Adj : Adj := Build_NatIso phi_iso.
Variables (d : D) (c : C) (ad : Adj).
Definition ApAphi (f : F d --> c) : d --> G c := ad (Build_POb1 d c) f.
Definition ApAphi_inv (g : d --> G c) : F d --> c :=
NatIso_inv ad (Build_POb1 d c) g.
(* rewrite rules *)

(* Specifications: *)
Lemma AphiPres : forall f g : F d --> c, f =_S g -> ApAphi f =_S ApAphi g.
Lemma Aphi_invPres :
forall f g : d --> G c, f =_S g -> ApAphi_inv f =_S ApAphi_inv g.