proposition: Lemma Tinf_colimIn_alg_map_commutes (v : vertex nat_graph) : T ⊗^{ V}_{l} colimIn free_monoid_coeq_sequence_colim_unit v · (Mon_alg_map _ (pr2 TinfM)) = free_monoid_coeq_sequence_on I_{ V} (v) · colimIn free_monoid_coeq_sequence_colim_unit (S v).
proof: Proof. etrans. apply assoc. apply (post_comp_with_z_iso_is_inj (pr2 Hconv)). etrans. apply assoc'. etrans. apply cancel_precomposition. exact (pr22 Hconv). etrans. apply id_right. apply pathsinv0. etrans. apply assoc. etrans. apply cancel_postcomposition. { etrans. apply assoc'. etrans. apply cancel_precomposition. etrans. apply assoc. etrans. apply cancel_postcomposition. exact (pathsinv0 (monoidal_leftunitorinvnat V _ _ _)). etrans. apply assoc'. apply cancel_precomposition. exact (pathsinv0 (whiskerscommutes V (bifunctor_equalwhiskers V) _ _)). etrans. apply assoc. apply assoc. } apply pathsinv0. etrans. apply cancel_postcomposition. { set (isHCC := HT _ (pr2 (free_monoid_coeq_sequence_colim_on I_{V}))). set (HCC := make_ColimCocone _ _ _ isHCC). etrans. { etrans. exact (pathsinv0 (id_left _)). etrans. apply cancel_postcomposition. exact (pathsinv0 (bifunctor_rightid V _ _)). exact (pathsinv0 (colimInCommutes HCC _ _ (idpath (S v)))). } etrans. apply cancel_postcomposition. { etrans. apply cancel_postcomposition. apply (bifunctor_rightid V). apply id_left. } etrans. apply cancel_precomposition. { etrans. apply cancel_postcomposition. apply (bifunctor_rightid V). apply id_left. } reflexivity. } etrans. apply assoc'. etrans. apply cancel_precomposition. exact (Tinf_limit_step_colimIn_commutes v). etrans. apply assoc. etrans. apply cancel_postcomposition. apply assoc. apply pathsinv0. etrans. apply assoc'. etrans. apply cancel_precomposition. exact (Tinf_limit_step_colimIn_commutes v). etrans. apply assoc. etrans. apply cancel_postcomposition. apply assoc. apply cancel_postcomposition. apply cancel_postcomposition. set (coeq := (next_pair_diagram_coeq (free_monoid_coeq_sequence_on I_{ V} v))). etrans. apply assoc'. etrans. apply assoc'. etrans. apply cancel_precomposition. apply assoc. etrans. apply assoc. exact (CoequalizerArrowEq _ coeq). 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