{A B : Type} (f : A -> B) : (forall (P : B -> Type) {P_inO : forall b:B, In O (P b)} (d : forall a:A, P (f a)), ExtensionAlong f P d) -> IsConnMap O f. Proof. intros Hf b. apply isconnected_from_elim_to_O. assert (e := Hf (fun b => O (hfiber f b)) _ (fun a => to O _ (a;1))). exists (e.1 b). intros [a p]. destruct p. symmetry; apply (e.2). 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.