{A B C : Type} (f : A -> B) (g : B -> C) (a : A) : ap (O_functor g) (to_O_natural f a) @ to_O_natural g (f a) = (O_functor_compose f g (to O A a))^ @ to_O_natural (g o f) a. Proof. unfold O_functor_compose, to_O_natural. rewrite O_indpaths_beta. rewrite !inv_pp, ap_V, !inv_V, !concat_pp_p. rewrite concat_Vp, concat_p1; reflexivity. Qed.
Require Import HoTT.Basics HoTT.Types. Require Import Equiv.BiInv Extensions HProp HFiber NullHomotopy Limits.Pullback. Require Import Homotopy.IdentitySystems. Require Import Colimits.Pushout.