+ 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.