+record category1 : Type2 ≝
+ { objs1:> Type;
+ arrows1: objs1 → objs1 → setoid1;
+ id1: ∀o:objs1. arrows1 o o;
+ comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
+ comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
+ comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
+ id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
+ id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
+ }.
+
+notation "'ASSOC'" with precedence 90 for @{'assoc}.
+notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
+
+interpretation "category1 composition" 'compose x y = (fun1 ??? (comp1 ????) y x).
+interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ????????).
+interpretation "category composition" 'compose x y = (fun ??? (comp ????) y x).
+interpretation "category assoc" 'assoc = (comp_assoc ????????).
+
+definition unary_morphism_setoid: setoid → setoid → setoid.
+ intros;
+ constructor 1;
+ [ apply (unary_morphism s s1);
+ | constructor 1;
+ [ intros (f g); apply (∀a. f a = g a);
+ | intros 1; simplify; intros; apply refl;
+ | simplify; intros; apply sym; apply H;
+ | simplify; intros; apply trans; [2: apply H; | skip | apply H1]]]
+qed.
+
+notation "hbox(a break ⇒ b)" right associative with precedence 20 for @{ 'Imply $a $b }.
+interpretation "unary morphism" 'Imply a b = (unary_morphism_setoid a b).
+interpretation "unary morphism" 'Imply a b = (unary_morphism a b).
+
+definition SET: category1.
+ constructor 1;
+ [ apply setoid;
+ | apply rule (λS,T.unary_morphism_setoid S T);
+ | intros; constructor 1; [ apply (λx.x); | intros; assumption ]
+ | intros; constructor 1; [ intros; constructor 1; [ apply (λx. c1 (c x)); | intros;
+ apply († (†H));]
+ | intros; whd; intros; simplify; whd in H1; whd in H;
+ apply trans; [ apply (b (a' a1)); | lapply (prop_1 ?? b (a a1) (a' a1));
+ [ apply Hletin | apply (H a1); ] | apply H1; ]]
+ | intros; whd; intros; simplify; apply refl;
+ | intros; simplify; whd; intros; simplify; apply refl;
+ | intros; simplify; whd; intros; simplify; apply refl;
+ ]
+qed.
+
+definition setoid_OF_SET: objs1 SET → setoid.
+ intros; apply o; qed.
+
+coercion setoid_OF_SET.
+
+
+definition prop_1_SET :
+ ∀A,B:SET.∀w:arrows1 SET A B.∀a,b:A.eq1 ? a b→eq1 ? (w a) (w b).
+intros; apply (prop_1 A B w a b H);
+qed.
+
+interpretation "SET dagger" 'prop1 h = (prop_1_SET ? ? ? ? ? h).