proposition: Lemma reflects_isos_cat_of_elems_forgetful : reflects_isos cat_of_elems_forgetful.
proof: 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.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Presheaf