Require Import Enrichment_ch2_8.
Require Import RepresentableStructure_ch7_2.
Require Import GeneralizedArrow.
+Require Import WeakFunctorCategory.
+Require Import SmallSMMEs.
-Definition CategoryOfGeneralizedArrows : Category SMME (fun x y => @GeneralizedArrow x _ y).
- admit.
- Qed.
+Inductive GeneralizedArrowOrIdentity : SMMEs -> SMMEs -> Type :=
+ | gaoi_id : forall smme:SMMEs, GeneralizedArrowOrIdentity smme smme
+ | gaoi_ga : forall s1 s2:SMMEs, GeneralizedArrow s1 s2 -> GeneralizedArrowOrIdentity s1 s2.
+
+Definition generalizedArrowOrIdentityFunc
+ : forall s1 s2, GeneralizedArrowOrIdentity s1 s2 -> { fobj : _ & Functor s1 s2 fobj }.
+ intros.
+ destruct X.
+ exists (fun x => x).
+ apply functor_id.
+ eapply existT.
+ apply (g >>>> RepresentableFunctor s2 (mon_i s2)).
+ Defined.
+
+Definition compose_generalizedArrows (s0 s1 s2:SMMEs) :
+ GeneralizedArrow s0 s1 -> GeneralizedArrow s1 s2 -> GeneralizedArrow s0 s2.
+ intro g01.
+ intro g12.
+ refine
+ {| ga_functor := g01 >>>> RepresentableFunctor s1 (mon_i s1) >>>> g12 |}.
+ apply MonoidalFunctorsCompose.
+ apply MonoidalFunctorsCompose.
+ apply (ga_functor_monoidal g01).
+ apply (me_mf s1).
+ apply (ga_functor_monoidal g12).
+ Defined.
+
+Definition generalizedArrowOrIdentityComp
+ : forall s1 s2 s3, GeneralizedArrowOrIdentity s1 s2 -> GeneralizedArrowOrIdentity s2 s3 -> GeneralizedArrowOrIdentity s1 s3.
+ intros.
+ destruct X.
+ apply X0.
+ destruct X0.
+ apply (gaoi_ga _ _ g).
+ apply (gaoi_ga _ _ (compose_generalizedArrows _ _ _ g g0)).
+ Defined.
+
+Definition MorphismsOfCategoryOfGeneralizedArrows : @SmallFunctors SMMEs.
+ refine {| small_func := GeneralizedArrowOrIdentity
+ ; small_func_id := fun s => gaoi_id s
+ ; small_func_func := fun smme1 smme2 f => projT2 (generalizedArrowOrIdentityFunc _ _ f)
+ ; small_func_comp := generalizedArrowOrIdentityComp
+ |}; intros; simpl.
+ apply if_id.
+ destruct f as [|fobj f]; simpl in *.
+ apply if_inv.
+ apply if_left_identity.
+ destruct g as [|gobj g]; simpl in *.
+ apply if_inv.
+ apply if_right_identity.
+ unfold mf_F.
+ idtac.
+ unfold mf_f.
+ apply if_associativity.
+ Defined.
+
+Definition CategoryOfGeneralizedArrows :=
+ WeakFunctorCategory MorphismsOfCategoryOfGeneralizedArrows.