proposition: Lemma free_monoid_coeq_sequence_colim_on_Tinf_Tinf_map_formscocone (m : vertex nat_graph) : pair_diagram_horizontal_arrow (free_monoid_coeq_sequence_on I_{V} m) ⊗^{V}_{r} Tinf · free_monoid_coeq_sequence_diagram_on_Tinf_Tinf_map (S m) = free_monoid_coeq_sequence_diagram_on_Tinf_Tinf_map m.
proof: Proof. set (pdTinfmap_rel := pr22 (free_monoid_coeq_sequence_on_Tinf_pd_Tinf_map m)). etrans. apply cancel_postcomposition. apply (bifunctor_rightcomp V). etrans. apply assoc'. etrans. apply cancel_precomposition. exact pdTinfmap_rel. etrans. apply cancel_postcomposition. apply (bifunctor_rightcomp V). etrans. apply assoc. etrans. apply cancel_postcomposition, assoc. etrans. apply assoc'. etrans. apply assoc4. etrans. apply cancel_postcomposition, cancel_precomposition. exact (pathsinv0 (monoidal_associatornatright V _ _ _ _ _)). etrans. apply cancel_postcomposition, assoc. etrans. apply assoc. etrans. apply assoc4. etrans. apply cancel_postcomposition, cancel_precomposition. apply (whiskerscommutes V (bifunctor_equalwhiskers V)). etrans. apply cancel_postcomposition, assoc. etrans. apply assoc'. etrans. apply cancel_precomposition. { etrans. exact (pathsinv0 (id_left _)). etrans. apply cancel_postcomposition. exact (pathsinv0 (pr1 (monoidal_leftunitorisolaw V _))). etrans. apply assoc'. etrans. apply cancel_precomposition. etrans. apply assoc. exact (pr22 TinfM). apply id_right. } etrans. apply assoc'. etrans. apply cancel_precomposition. apply (monoidal_leftunitornat V). etrans. apply assoc. etrans. apply cancel_postcomposition. { etrans. do 2 apply cancel_postcomposition. exact (pathsinv0 (monoidal_triangle_identity'_inv V _ _)). etrans. apply assoc4. etrans. apply cancel_postcomposition, cancel_precomposition. apply (monoidal_associatorisolaw V). etrans. apply cancel_postcomposition. apply id_right. apply (monoidal_leftunitorisolaw V). } apply id_left. 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