Definition functor_op_of_postcomp_functor_factorizes_through_functorcatofopp_as_nat_trans : nat_trans (functor_op (post_composition_functor A B C F)) (functor_composite (opfunctorcat_to_functorcatofoppcats A B) (functor_composite (post_composition_functor _ _ _ (functor_op F)) (opfunctorcat_from_functorcatofoppcats A C)) ). Proof. use make_nat_trans. - intro. use make_nat_trans. + intro ; apply identity. + abstract (intro ; intros ; apply (id_right _ @ ! id_left _)). - abstract (intro ; intros ; use nat_trans_eq ; [ apply homset_property | intro ; 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.