{A B : Type} (f g : A -> B) (pi : f == g) (x : A) : O_functor_homotopy f g pi (to O A x) = to_O_natural f x @ ap (to O B) (pi x) @ (to_O_natural g x)^. Proof. unfold O_functor_homotopy, to_O_natural. refine (O_indpaths_beta _ _ _ x @ _). refine (concat_p_pp _ _ _). 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.