proposition: Lemma Tinf_monoid_unit_right_pointwise (v : vertex nat_graph) : ruinv^{ V }_{ pair_diagram_lob (free_monoid_coeq_sequence_on I_{ V} v)} · (pair_diagram_lob (free_monoid_coeq_sequence_on I_{ V} v) ⊗^{ V}_{l} colimIn free_monoid_coeq_sequence_colim_unit 0 · free_monoid_coeq_sequence_diagram_on_Tinf_Tinf_map v) = colimIn free_monoid_coeq_sequence_colim_unit v.
proof: Proof. induction v as [|v IHv]. { etrans. apply cancel_precomposition. apply (monoidal_leftunitornat V). etrans. apply assoc. etrans. do 2 apply cancel_postcomposition. exact (pathsinv0 (unitorsinv_coincide_on_unit V)). etrans. apply cancel_postcomposition. apply (monoidal_leftunitorisolaw V). apply id_left. } induction v as [|v _]. - etrans. apply cancel_precomposition. { etrans. apply assoc. apply cancel_postcomposition. exact (pathsinv0 (whiskerscommutes V (bifunctor_equalwhiskers V) _ _)). } etrans. apply assoc. etrans. apply cancel_postcomposition. etrans. apply assoc. etrans. apply cancel_postcomposition. etrans. apply (monoidal_rightunitorinvnat V). exact (pr1 (monoidal_rightunitorisolaw V _)). apply id_left. etrans. exact (Tinf_colimIn_alg_map_commutes 0). apply id_left. - set (Hn := free_monoid_coeq_sequence_on_Tinf_pd_Tinf_map v). set (τn1 := pr12 Hn). set (coeq := next_pair_diagram_coeq (free_monoid_coeq_sequence_on I_{V} v)). use CoequalizerOutUnique'. etrans. apply assoc. etrans. apply cancel_postcomposition. exact (pathsinv0 (monoidal_rightunitorinvnat V _ _ _)). etrans. apply assoc'. etrans. apply cancel_precomposition. etrans. apply assoc. etrans. apply cancel_postcomposition. apply (whiskerscommutes V (bifunctor_equalwhiskers V)). etrans. apply assoc'. apply cancel_precomposition. exact (pr22 (free_monoid_coeq_sequence_on_Tinf_pd_Tinf_map (S v))). etrans. apply cancel_precomposition. etrans. apply assoc. apply cancel_postcomposition. etrans. apply assoc. apply cancel_postcomposition. exact (pathsinv0 (monoidal_associatornatleft V _ _ _ _ _)). etrans. apply cancel_postcomposition. exact (pathsinv0 (monoidal_triangle_identity''_inv V _ _)). etrans. apply assoc'. etrans. apply cancel_precomposition. etrans. apply assoc. apply cancel_postcomposition. etrans. apply assoc. apply cancel_postcomposition. etrans. apply assoc. etrans. apply cancel_postcomposition. apply (monoidal_associatorisolaw V). apply id_left. etrans. apply assoc. etrans. apply cancel_postcomposition. etrans. apply cancel_precomposition. exact (pathsinv0 (bifunctor_leftcomp V _ _ _ _ _ _)). etrans. exact (pathsinv0 (bifunctor_leftcomp V _ _ _ _ _ _)). apply maponpaths. exact IHv. exact (Tinf_colimIn_alg_map_commutes (S 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