(c : C) : equiv_O_pushout (to O (Pushout f g) (pushr c)) = to O (Pushout (O_functor f) (O_functor g)) (pushr (to O C c)). Proof. cbn. rapply to_O_natural. 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.