proposition: Lemma Tinf_monoid_assoc : monoid_laws_assoc V Tinf_monoid_data.
proof: Proof. set (ccTinf := free_monoid_coeq_sequence_colim_on I_{V}). set (rt_ccTinf := rt_chain (Tinf ⊗_{V} Tinf) _ _ _ (isColimCocone_from_ColimCocone ccTinf)). set (rt_colim := free_monoid_coeq_sequence_rightwhisker_colim_on (Tinf ⊗_{V} Tinf) I_{V}). set (iso := (_,, isColim_is_z_iso _ rt_colim _ _ rt_ccTinf) : z_iso _ _). apply (pre_comp_with_z_iso_is_inj (is_inverse_in_precat_inv (monoidal_associatorisolaw V _ _ _))). apply (pre_comp_with_z_iso_is_inj iso). use colimArrowUnique'. intro v. etrans. apply assoc. etrans. apply cancel_postcomposition. apply colimArrowCommutes. etrans. apply cancel_precomposition. { etrans. apply assoc. apply cancel_postcomposition. etrans. apply assoc. etrans. apply cancel_postcomposition. apply (monoidal_associatorisolaw V). apply id_left. } etrans. apply cancel_postcomposition. etrans. apply cancel_precomposition. apply (bifunctor_leftid V). apply id_right. etrans. apply assoc. etrans. apply cancel_postcomposition. apply (whiskerscommutes V (bifunctor_equalwhiskers V)). etrans. apply assoc'. etrans. apply cancel_precomposition. etrans. apply assoc. etrans. apply cancel_postcomposition. exact (Tinf_monoid_mul_iso_precomp_with_colimIn v). apply (colimArrowCommutes ((free_monoid_coeq_sequence_rightwhisker_colim_on Tinf I_{ V}))). apply pathsinv0. etrans. apply assoc. etrans. apply cancel_postcomposition. apply colimArrowCommutes. etrans. apply cancel_postcomposition. etrans. apply cancel_precomposition. apply (bifunctor_leftid V). apply id_right. etrans. apply assoc. etrans. apply cancel_postcomposition. apply (monoidal_associatorinvnatright V). etrans. apply assoc'. etrans. apply cancel_precomposition. { etrans. apply assoc. etrans. apply cancel_postcomposition. { etrans. exact (pathsinv0 (bifunctor_rightcomp V _ _ _ _ _ _)). apply maponpaths. etrans. apply assoc. etrans. apply cancel_postcomposition. exact (Tinf_monoid_mul_iso_precomp_with_colimIn v). apply (colimArrowCommutes ((free_monoid_coeq_sequence_rightwhisker_colim_on Tinf I_{ V}))). } reflexivity. } etrans. apply assoc. exact (Tinf_monoid_assoc_pointwise v). Qed.
imports: 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.Limits.Graphs.Coequalizers UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Chains.Chains UniMath.Combinatorics.StandardFiniteSets UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.CategoriesOfMonoids UniMath.ModelCategories.Generated.GenericFreeMonoid UniMath.ModelCategories.Generated.MonoidalHelpers UniMath.ModelCategories.Helpers