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(* 02110-1301 USA                                                     *)
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(*          Projet Coq - Calculus of Inductive Constructions V5.10           *)
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(*                                                                           *)
(*          Category Theory : Natural Transformations                        *)
(*                                                                           *)
(*          Amokrane Saibi May 1994                                          *)
(*                                                                           *)
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Require Export Functor.
Require Export Setoid_dup2.
Set Implicit Arguments.
Unset Strict Implicit.
(* Natural Transformations *)
Section nt_def.
Variables (C D : Category) (F G : Functor C D).
Definition NT_law (T : forall a : C, F a --> G a) :=
forall (a b : C) (f : a --> b), FMor F f o T b =_S T a o FMor G f.
Structure > NT : Type :=
{ApNT :> forall a : C, F a --> G a; Prf_NT_law :> NT_law ApNT}.
(*** rewrite rules ***)

(* Specifications: *)
Lemma NatCond :
forall (T : NT) (a b : C) (f : a --> b), FMor F f o T b =_S T a o FMor G f.
Lemma NatCond1 :
forall (T : NT) (a b : C) (f : a --> b), T a o FMor G f =_S FMor F f o T b.