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(* 02110-1301 USA                                                     *)
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(*          Projet Formel - Calculus of Inductive Constructions V5.10        *)
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(*									     *)
(*	                Adjunction Example : Cartesian         		     *)
(*									     *)
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(*									     *)
(*                     A. SAIBI	  May 95                  		     *)
(*									     *)
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Require Export Diagonal.
Require Export CCC.
Require Export Terminal1.
Set Implicit Arguments.
Unset Strict Implicit.
(* Diag(C) a un RAdj -> C admet les produits *)
Section diag_hasprod1.
Variable C : Category.
SubClass HasBinProd1 := RightAdj (Diag C).
Structure IsCartesian1 : Type :=
{Car_terminal1 :> Terminal1 C; Car_BP1 :> HasBinProd1}.
Variable H : HasBinProd1.
Section diag_prod1.
Variable a b : C.
Let ua' := CoUnit H (Build_POb a b).
Definition Obj_prod1 := CoUA_ob ua'.
Definition Proj1_prod1 := Hom_l (CoUA_mor ua').
Definition Proj2_prod1 := Hom_r (CoUA_mor ua').
Section together1_def.
Variable c : C.
Definition Together1 (f : c --> a) (g : c --> b) :=
CoUA_diese ua' (Build_Pmor (u:=Build_POb c c) (t:=Build_POb a b) f g).

(* Specifications: *)
Lemma Together1_l : Map2_congl_law Together1.
Lemma Together1_r : Map2_congr_law Together1.
Definition Op_together1 := Build_Map2 Together1_l Together1_r.