{A B C : Type} (f : A -> B) (g : B -> C) : MapIn O g -> MapIn O (g o f) -> MapIn O f. Proof. intros ? ? b. refine (inO_equiv_inO _ (hfiber_hfiber_compose_map f g b)). Defined.
Require Import HoTT.Basics HoTT.Types. Require Import Equiv.BiInv Extensions HProp HFiber NullHomotopy Limits.Pullback. Require Import Homotopy.IdentitySystems. Require Import Colimits.Pushout.