{A B : Type} `{In O A} `{In O B} (f : A -> B) : (to O B)^-1 o (O_functor f) == f o (to O A)^-1. Proof. refine (O_indpaths _ _ _); intros x. apply moveR_equiv_V. refine (to_O_natural f x @ _). do 2 apply ap. symmetry; apply eissect. 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.