{A B} (f : A <~> B) : (equiv_O_functor f) o (to O A) == (to O B) o f := to_O_natural f.
Require Import HoTT.Basics HoTT.Types. Require Import Equiv.BiInv Extensions HProp HFiber NullHomotopy Limits.Pullback. Require Import Homotopy.IdentitySystems. Require Import Colimits.Pushout.