proposition: Proposition sym_mon_hexagon_lassociator1 (x y1 y2 z : V) : mon_lassociator x (y1 ⊗ y2) z · x ⊗^{ V}_{l} (mon_lassociator y1 y2 z · y1 ⊗^{ V}_{l} sym_mon_braiding y2 z) · mon_rassociator x y1 (z ⊗ y2) = mon_rassociator x y1 y2 ⊗^{ V}_{r} z · mon_lassociator (x ⊗ y1) y2 z · (x ⊗ y1) ⊗^{ V}_{l} sym_mon_braiding y2 z.
proof: Proof. apply pathsinv0. use (z_iso_inv_on_left _ _ _ _ (mon_lassociator _ _ _ ,, mon_rassociator _ _ _ ,, _)). { apply monoidal_associatorisolaw. } cbn. etrans. 2: { rewrite assoc'. apply maponpaths. apply monoidal_associatornatleft. } rewrite (bifunctor_leftcomp V). rewrite ! assoc. apply maponpaths_2. etrans. 2: { rewrite assoc'. apply maponpaths. apply pathsinv0, mon_lassociator_lassociator. } rewrite ! assoc. rewrite <- (when_bifunctor_becomes_leftwhiskering V). apply maponpaths_2. rewrite <- (when_bifunctor_becomes_rightwhiskering V). etrans. 2: { apply maponpaths_2. apply tensor_comp_id_r. } rewrite mon_rassociator_lassociator. rewrite tensor_id_id. exact (! id_left _). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories