-interpretation "category1 composition" 'compose x y = (fun1 ___ (comp1 ____) x y).
-interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
-interpretation "category composition" 'compose x y = (fun ___ (comp ____) x y).
-interpretation "category assoc" 'assoc = (comp_assoc ________).
+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).