+Require Import GeneralizedArrow.
+
+Section ExtendFunctor.
+
+ Context `(F:Functor).
+ Context (P:c1 -> Prop).
+
+ Definition domain_subcat := FullSubcategory c1 P.
+
+ Definition functor_restricts_to_full_subcat_on_domain_fobj (a:domain_subcat) : c2 :=
+ F (projT1 a).
+
+ Definition functor_restricts_to_full_subcat_on_domain_fmor (a b:domain_subcat)(f:a~~{domain_subcat}~~>b) :
+ (functor_restricts_to_full_subcat_on_domain_fobj a)~~{c2}~~>(functor_restricts_to_full_subcat_on_domain_fobj b) :=
+ F \ (projT1 f).
+
+ Lemma functor_restricts_to_full_subcat_on_domain : Functor domain_subcat c2 functor_restricts_to_full_subcat_on_domain_fobj.
+ refine {| fmor := functor_restricts_to_full_subcat_on_domain_fmor |};
+ unfold functor_restricts_to_full_subcat_on_domain_fmor; simpl; intros.
+ setoid_rewrite H; reflexivity.
+ setoid_rewrite fmor_preserves_id; reflexivity.
+ setoid_rewrite <- fmor_preserves_comp; reflexivity.
+ Defined.
+
+End ExtendFunctor.
+
+Section MonoidalSubCat.
+
+ (* a monoidal subcategory is a full subcategory, closed under tensor and containing the unit object *)
+ Class MonoidalSubCat {Ob}{Hom}{C:Category Ob Hom}{MFobj}{MF}{MI}(MC:MonoidalCat C MFobj MF MI) :=
+ { msc_P : MC -> Prop
+ ; msc_closed_under_tensor : forall o1 o2, msc_P o1 -> msc_P o2 -> msc_P (MC (pair_obj o1 o2))
+ ; msc_contains_unit : msc_P (mon_i MC)
+ ; msc_subcat := FullSubcategory MC msc_P
+ }.
+ Local Coercion msc_subcat : MonoidalSubCat >-> SubCategory.
+
+ Context `(MSC:MonoidalSubCat).
+
+ (* any full subcategory of a monoidal category, , is itself monoidal *)
+ Definition mf_restricts_to_full_subcat_on_domain_fobj (a:MSC ×× MSC) : MSC.
+ destruct a.
+ destruct o.
+ destruct o0.
+ set (MC (pair_obj x x0)) as m'.
+ exists m'.
+ apply msc_closed_under_tensor; auto.
+ Defined.
+
+ Definition mf_restricts_to_full_subcat_on_domain_fmor
+ {a}{b}
+ (f:a~~{MSC ×× MSC}~~>b)
+ :
+ (mf_restricts_to_full_subcat_on_domain_fobj a)~~{MSC}~~>(mf_restricts_to_full_subcat_on_domain_fobj b).
+ destruct a as [[a1 a1pf] [a2 a2pf]].
+ destruct b as [[b1 b1pf] [b2 b2pf]].
+ destruct f as [[f1 f1pf] [f2 f2pf]].
+ simpl in *.
+ exists (MC \ (pair_mor (pair_obj a1 a2) (pair_obj b1 b2) f1 f2)); auto.
+ Defined.
+
+ Lemma mf_restricts_to_full_subcat_on_domain : Functor (MSC ×× MSC) MSC
+ mf_restricts_to_full_subcat_on_domain_fobj.
+ refine {| fmor := fun a b f => mf_restricts_to_full_subcat_on_domain_fmor f |};
+ unfold functor_restricts_to_full_subcat_on_domain_fmor; simpl; intros.
+ admit.
+ admit.
+ admit.
+ Defined.
+
+ Definition subcat_i : MSC.
+ exists (mon_i MC).
+ apply msc_contains_unit.
+ Defined.
+
+ Lemma full_subcat_is_monoidal : MonoidalCat MSC _ mf_restricts_to_full_subcat_on_domain subcat_i.
+ admit.
+ Defined.
+
+ Lemma inclusion_functor_monoidal : MonoidalFunctor full_subcat_is_monoidal MC (InclusionFunctor _ MSC).
+ admit.
+ Defined.
+
+End MonoidalSubCat.
+Coercion full_subcat_is_monoidal : MonoidalSubCat >-> MonoidalCat.
+