`{Funext} (A : pType) : IsCohHSpace A <~> { act : forall a, A ->* [A,a] & act pt ==* pmap_idmap }. Proof. refine (_ oE (issig_iscohhspace A)^-1). unfold IsPointed. equiv_via {s : {act : A -> (A -> A) & forall a, act a pt = a} & {h : s.1 pt == idmap & h pt = s.2 pt}}. 1: make_equiv. snrapply equiv_functor_sigma'. - refine (equiv_functor_forall_id (fun a => issig_pmap A [A,a]) oE _). unfold IsPointed. nrapply equiv_sig_coind. - cbn. intros [act p]; simpl. refine (issig_phomotopy _ _ oE _); cbn. apply equiv_functor_sigma_id; intro q. apply equiv_concat_r; symmetry; apply concat_p1. Defined.
Require Import Basics Types HSpace.Core HSpace.Coherent HSpace.Pointwise