`{Funext} {A : Type} {P Q : A -> Type} (f : forall a, P a -> Q a) `{forall a, MapIn O (f a)} : MapIn O (functor_forall_id f). Proof. intro g. rapply (inO_equiv_inO _ (hfiber_functor_forall_id f g)^-1%equiv). 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.