proposition: Lemma unassoc_commutes : nat_z_iso (HHH ∙ (precategory_binproduct_unassoc D D D)) ((precategory_binproduct_unassoc C C C) ∙ (pair_functor H HH)).
proof: Proof. use make_nat_z_iso. - use make_nat_trans. + intro ; use catbinprodmor ; apply identity. + intro ; intros. use total2_paths_f. * exact (id_right _ @ ! id_left _). * abstract (rewrite transportf_const ; exact (id_right _ @ ! id_left _)). - intro. use tpair. * use catbinprodmor ; apply identity. * abstract (split ; (use total2_paths_f ; [ apply id_right | rewrite transportf_const ; apply id_right ])). Defined.
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.Core.Univalence UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.catiso UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.PrecompEquivalence UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorCategory UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.TotalCategoryFacts UniMath.CategoryTheory.Monoidal.RezkCompletion.LiftedTensor UniMath.CategoryTheory.Monoidal.RezkCompletion.LiftedTensorUnit