O (Pushout f g) <~> O (Pushout (O_functor f) (O_functor g)) := Build_Equiv _ _ _ (O_inverts_functor_pushout (to O A) (to O B) (to O C) (fun x => (to_O_natural f x)^) (fun x => (to_O_natural g x)^)).
Require Import HoTT.Basics HoTT.Types. Require Import Equiv.BiInv Extensions HProp HFiber NullHomotopy Limits.Pullback. Require Import Homotopy.IdentitySystems. Require Import Colimits.Pushout.