Definition adj_equivalence_ptdfunctor_in_cat : adjoint_equivalence (B := bicat_of_cats) (coslice_cat_total V (monoidal_unit Mon_V)) (coslice_cat_total (coslice_cat_total V (monoidal_unit Mon_V)) (monoidal_unit (monoidal_pointed_objects Mon_V))). Proof. use make_adjoint_equivalence. - exact (ptdob_to_ptdptdob Mon_V). - exact (ptdptdob_to_ptdob Mon_V). - exact (unit_ptdob Mon_V). - exact (counit_ptdob Mon_V). - use nat_trans_eq. { apply homset_property. } intro. use total2_paths_f. 2: { apply homset_property. } use total2_paths_f. 2: { apply homset_property. } simpl. rewrite ! id_right. apply idpath. - use nat_trans_eq. { apply homset_property. } intro. use total2_paths_f. 2: { apply homset_property. } simpl. rewrite ! id_right. apply idpath. - repeat (use tpair). + intro. exists (identity _). apply id_right. + intro ; intros. use total2_paths_f. 2: { apply homset_property. } etrans. { apply id_right. } apply (! id_left _). + apply nat_trans_eq. { apply homset_property. } intro. use total2_paths_f. 2: { apply homset_property. } apply id_right. + apply nat_trans_eq. { apply homset_property. } intro. use total2_paths_f. 2: { apply homset_property. } apply id_right. - repeat (use tpair). + intro. exists (identity _). use total2_paths_f. 2: { apply homset_property. } simpl. rewrite (! pr22 x). rewrite id_left. apply id_right. + intro ; intros. use total2_paths_f. 2: { apply homset_property. } etrans. { apply id_right. } apply (! id_left _). + apply nat_trans_eq. { apply homset_property. } intro. use total2_paths_f. 2: { apply homset_property. } apply id_right. + apply nat_trans_eq. { apply homset_property. } intro. use total2_paths_f. 2: { apply homset_property. } apply id_right. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Functors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors. Require Import UniMath.CategoryTheory.Monoidal.Displayed.Monoidal. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Require Import UniMath.Bicategories.MonoidalCategories.PointedFunctorsWhiskeredMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.EndofunctorsWhiskeredMonoidal. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.Bicategories.MonoidalCategories.BicatOfWhiskeredMonCats. Require Import UniMath.Bicategories.DisplayedBicats.DispAdjunctions.