Lemma reflects_isos_cat_of_elems_forgetful : reflects_isos cat_of_elems_forgetful. Proof. intros [c x] [d y] f Hf. destruct f as [f i]; destruct Hf as [f' j]. assert (i' : y = #X f' x). { intermediate_path (#X (identity d) y). - exact (eqtohomot (!functor_id X d) y). - intermediate_path (#X (f ∘ f') y). + exact (eqtohomot (!maponpaths #X (pr2 j)) y). + intermediate_path (#X f' (#X f y)). * exact (eqtohomot ((functor_comp X) f f') y). * exact (maponpaths (#X f') (!i)). } exists (f',,i'). split; apply cat_of_elems_mor_eq; [ exact (pr1 j) | exact (pr2 j) ]. Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Presheaf.