proposition: Lemma slicecat_functor_identity (x : C) : slicecat_functor (identity x) = functor_identity (C / x).
proof: Proof. apply (functor_eq _ _ (has_homsets_slice_precat _ _)); simpl. apply (two_arg_paths_f (slicecat_functor_identity_ob _)). apply funextsec; intros [a f]. apply funextsec; intros [b g]. apply funextsec; intros [h hh]. rewrite transport_of_functor_map_is_pointwise; simpl in *. unfold double_transport. unfold slicecat_mor, slice_precat_ob_mor, slicecat_mor. simpl. rewrite transportf_total2. apply subtypePairEquality; [intro; apply C | ]. rewrite transportf_total2; simpl. unfold slicecat_functor_identity_ob. rewrite toforallpaths_funextsec; simpl. induction (id_right f). induction (id_right g). apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Univalence UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis