proposition: Proposition indexed_cat_to_disp_cat_to_disp_iso_inv_right : indexed_cat_to_disp_cat_to_disp_iso_inv · # (Φ $ identity x) (f · indexed_cat_id Φ x xx₂) · indexed_cat_comp Φ (identity x) (identity x) xx₂ = indexed_cat_id Φ x xx₂ · idtoiso (maponpaths (λ h, (Φ $ h) xx₂) (!(id_left _))).
proof: Proof. unfold indexed_cat_to_disp_cat_to_disp_iso_inv. rewrite functor_comp. rewrite !assoc'. etrans. { do 3 apply maponpaths. apply (indexed_cat_lunitor_alt Φ). } etrans. { apply maponpaths. rewrite !assoc. apply maponpaths_2. refine (!_). apply (nat_trans_ax (indexed_cat_id Φ x)). } cbn. rewrite !assoc. rewrite z_iso_after_z_iso_inv. refine (maponpaths (λ z, z · _) (id_left _) @ _). do 4 apply maponpaths. apply homset_property. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.OppositeCategory.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.IndexedCategories.IndexedCategory