proposition: Lemma total_adjunction_forms_equivalence2_disp_inv_is_inv : ∏ x : total_category D2, is_disp_inverse (z_iso_is_inverse_in_precat (pr1 (adjcounit (total_adjunction FF) x),, total_adjunction_forms_equivalence2_base x)) (pr2 (adjcounit (total_adjunction FF) x)) (total_adjunction_forms_equivalence2_disp_inv x).
proof: Proof. unfold total_adjunction_forms_equivalence2_base. unfold total_adjunction_forms_equivalence2_disp_inv. split. - cbn. use transportf_transpose_right. unfold transportb. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. cbn. set (ff := pr12 (pr22 FF (pr1 x) (pr2 x))). cbn in ff. set (fff := transportf_transpose_left ff). refine (_ @ fff). use transportf_transpose_right. unfold transportb. rewrite transport_f_f. rewrite id_right_disp. rewrite id_left_disp. rewrite transport_b_b. unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. use transportf_set. apply homset_property. - cbn. use transportf_transpose_right. unfold transportb. rewrite mor_disp_transportf_prewhisker. rewrite transport_f_f. cbn. set (ff := pr22 (pr22 FF (pr1 x) (pr2 x))). cbn in ff. set (fff := transportf_transpose_left ff). refine (_ @ fff). use transportf_transpose_right. unfold transportb. rewrite transport_f_f. rewrite id_right_disp. rewrite id_left_disp. rewrite transport_b_b. unfold transportb. rewrite mor_disp_transportf_postwhisker. rewrite transport_f_f. use transportf_set. apply homset_property. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Adjunctions UniMath.CategoryTheory.DisplayedCats.Equivalences UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Isos