proposition: Lemma monoidal_triangle_identity_inv {C : category} (M : monoidal C) (x y : C) : x ⊗^{M}_{l} luinv_{M} y · αinv_{M} x I_{ M} y = ruinv_{M} x ⊗^{ M}_{r} y.
proof: Proof. apply pathsinv0. apply (z_iso_inv_on_left _ _ _ _ ((z_iso_from_associator_iso M _ _ _))). cbn. set (luiy := make_z_iso _ _ (monoidal_leftunitorisolaw M y)). set (luixy := functor_on_z_iso (leftwhiskering_functor M x) luiy). set (ruix := make_z_iso _ _ (monoidal_rightunitorisolaw M x)). set (ruixy := functor_on_z_iso (rightwhiskering_functor M y) ruix). apply pathsinv0. apply (z_iso_inv_on_right _ _ _ ruixy). apply (z_iso_inv_on_left _ _ _ _ luixy). exact (! (monoidal_triangleidentity M) x y). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.Notations UniMath.MoreFoundations.PartA UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.FullyFaithful UniMath.CategoryTheory.opp_precat