(b : B) : equiv_O_pushout (to O (Pushout f g) (pushl b)) = to O (Pushout (O_functor f) (O_functor g)) (pushl (to O B b)). 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.