Generalizable All Variables.
Require Import Preamble.
Require Import General.
-Require Import NaturalDeduction.
-Require Import Coq.Strings.String.
-Require Import Coq.Lists.List.
+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 MonoidalCategories_ch7_8.
+Require Import Coherence_ch7_8.
+Require Import Enrichment_ch2_8.
+Require Import RepresentableStructure_ch7_2.
+Require Import Reification.
+Require Import WeakFunctorCategory.
+Require Import SmallSMMEs.
+
+Inductive ReificationOrIdentity : SMMEs -> SMMEs -> Type :=
+ | roi_id : forall smme:SMMEs, ReificationOrIdentity smme smme
+ | roi_reif : forall s1 s2:SMMEs, Reification s1 s2 (mon_i s2) -> ReificationOrIdentity s1 s2.
+
+Definition reificationOrIdentityFunc
+ : forall s1 s2, ReificationOrIdentity s1 s2 -> { fobj : _ & Functor s1 s2 fobj }.
+ intros.
+ destruct X.
+ exists (fun x => x).
+ apply functor_id.
+ exists (reification_rstar_obj r).
+ apply r.
+ Defined.
+
+Definition compose_reifications (s0 s1 s2:SMMEs) :
+ Reification s0 s1 (mon_i s1) -> Reification s1 s2 (mon_i s2) -> Reification s0 s2 (mon_i s2).
+ intros.
+ refine
+ {| reification_rstar := MonoidalFunctorsCompose _ _ _ _ _ (reification_rstar X) (reification_rstar X0)
+ ; reification_r := fun K => (reification_r X K) >>>> (reification_r X0 (mon_i s1))
+ |}.
+ intro K.
+ set (ni_associativity (reification_r X K) (reification_r X0 (mon_i s1)) (RepresentableFunctor s2 (mon_i s2))) as q.
+ eapply ni_comp.
+ apply q.
+ clear q.
+ set (reification_commutes X K) as comm1.
+ set (reification_commutes X0 (mon_i s1)) as comm2.
+ admit.
+ Defined.
+
+Definition reificationOrIdentityComp
+ : forall s1 s2 s3, ReificationOrIdentity s1 s2 -> ReificationOrIdentity s2 s3 -> ReificationOrIdentity s1 s3.
+ intros.
+ destruct X.
+ apply X0.
+ destruct X0.
+ apply (roi_reif _ _ r).
+ apply (roi_reif _ _ (compose_reifications _ _ _ r r0)).
+ Defined.
+
+Definition MorphismsOfCategoryOfReifications : @SmallFunctors SMMEs.
+ refine {| small_func := ReificationOrIdentity
+ ; small_func_id := fun s => roi_id s
+ ; small_func_func := fun smme1 smme2 f => projT2 (reificationOrIdentityFunc _ _ f)
+ ; small_func_comp := reificationOrIdentityComp
+ |}; 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.
+ apply if_id.
+ Defined.
+
+Definition CategoryOfReifications :=
+ WeakFunctorCategory MorphismsOfCategoryOfReifications.