Definition arrow_category_equiv_comma_category : adj_equivalence_of_cats arrow_category_to_comma_category. Proof. use make_adj_equivalence_of_cats. - exact comma_category_to_arrow_category. - exact arrow_category_equiv_comma_category_unit. - exact arrow_category_equiv_comma_category_counit. - exact arrow_category_comma_category_adjunction. - split. + intro ; cbn. use is_z_iso_arrow. * apply identity_is_z_iso. * apply identity_is_z_iso. + intro ; cbn. use is_z_iso_comma. * apply identity_is_z_iso. * apply identity_is_z_iso. 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.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.CommaCategories. Require Import UniMath.CategoryTheory.Adjunctions.Core.