proposition: Lemma functor_ass_to_nat_trans_ass {αC : associator TC} {αD : associator TD} {F : functor C D} {FF : functor_tensorunit_disp_cat TC TD IC ID F} (FF_iso : is_nat_z_iso (pr11 FF)) (FFF : functor_ass_disp_cat αC αD (_,,FF)) : functor_nat_trans_preserves αC αD FF_iso.
proof: Proof. use nat_trans_eq. { apply homset_property. } intro x. set (p := FFF (pr11 x) (pr21 x) (pr2 x)). simpl. rewrite assoc. rewrite ! (functor_id TD). rewrite ! id_left. rewrite ! id_right. refine (p @ _). apply assoc'. Qed.
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.whiskering UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored