+Require Import Categories_ch1_3.
+Require Import Functors_ch1_4.
+Require Import Isomorphisms_ch1_5.
+Require Import ProductCategories_ch1_6_1.
+Require Import OppositeCategories_ch1_6_2.
+Require Import Enrichment_ch2_8.
+Require Import Subcategories_ch7_1.
+Require Import NaturalTransformations_ch7_4.
+Require Import NaturalIsomorphisms_ch7_5.
+Require Import BinoidalCategories.
+Require Import PreMonoidalCategories.
+Require Import MonoidalCategories_ch7_8.
+Require Import Coherence_ch7_8.
+Require Import Enrichment_ch2_8.
+Require Import Enrichments.
+Require Import RepresentableStructure_ch7_2.
+Require Import GeneralizedArrow.
+Require Import WeakFunctorCategory.
+
+(*
+ * Technically reifications form merely a *semicategory* (no identity
+ * maps), but one can always freely adjoin identity maps (and nothing
+ * else) to a semicategory to get a category whose non-identity-map
+ * portion is identical to the original semicategory
+ *
+ * Also, technically this category has ALL enrichments (not just the
+ * surjective monic monoidal ones), though there maps OUT OF only the
+ * surjective enrichments and INTO only the monic monoidal
+ * enrichments. It's a big pain to do this in Coq, but sort of might
+ * matter in real life: a language with really severe substructural
+ * restrictions might fail to be monoidally enriched, meaning we can't
+ * use it as a host language. But that's for the next paper...
+ *)
+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 generalizedArrowOrIdentityFobj (s1 s2:SMMEs) (f:GeneralizedArrowOrIdentity s1 s2) : s1 -> s2 :=
+ match f in GeneralizedArrowOrIdentity S1 S2 return S1 -> S2 with
+ | gaoi_id s => fun x => x
+ | gaoi_ga s1 s2 f => fun a => ehom(ECategory:=s2) (enr_c_i (smme_e s2)) (ga_functor_obj f a)
+ end.
+
+Definition generalizedArrowOrIdentityFunc s1 s2 (f:GeneralizedArrowOrIdentity s1 s2)
+ : Functor s1 s2 (generalizedArrowOrIdentityFobj _ _ f) :=
+ match f with
+ | gaoi_id s => functor_id _
+ | gaoi_ga s1 s2 f => ga_functor f >>>> HomFunctor s2 (enr_c_i s2)
+ end.
+
+Instance compose_generalizedArrows (s0 s1 s2:SMMEs)
+ (g01:GeneralizedArrow s0 s1)(g12:GeneralizedArrow s1 s2) : GeneralizedArrow s0 s2 :=
+ { ga_functor_monoidal := g01 >>⊗>> smme_mon s1 >>⊗>> g12 }.
+ apply ga_host_lang_pure.
+ 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 => 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.
+ apply if_associativity.
+ Defined.
+
+Definition CategoryOfGeneralizedArrows :=
+ WeakFunctorCategory MorphismsOfCategoryOfGeneralizedArrows.