{A B : Type} (f : A -> B) `{In O A} `{In O B} `{IsConnMap O _ _ f} : IsEquiv f := isequiv_commsq' f (O_functor O f) (to O A) (to O B) (to_O_natural O f). Global Instance conn_map_O_functor {A B} (f : A -> B) `{IsConnMap O _ _ f} : IsConnMap O (O_functor O f). Proof. unfold O_functor. rapply conn_map_compose. 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.