Lemma counit_opfunctorcat_functorcatofoppcats : nat_trans (functor_composite (opfunctorcat_to_functorcatofoppcats C D) (opfunctorcat_from_functorcatofoppcats C D) ) (functor_identity ([C, D]^opp)). Proof. use make_nat_trans. - intro. use make_nat_trans. + intro ; apply identity. + abstract (intro ; intros ; cbn ; apply (id_right _ @ ! id_left _)). - abstract ( intro ; intros ; use nat_trans_eq ; [ apply homset_property | intro ; simpl ; apply (id_left _ @ ! 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.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.whiskering. Section OppositeOfFunctorCatFromFunctorCatOfOpposite. End OppositeOfFunctorCatFromFunctorCatOfOpposite.