(a : A) : equiv_O_coeq (to O (Coeq f g) (coeq a)) = to O (Coeq (O_functor f) (O_functor g)) (coeq (to O A a)). Proof. refine (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.