proposition: Lemma functor_comp_id_lax_specialized (F F' : C ⟶ D') (α: F ⟹ F') (β: F' ⟹ F) : nat_trans_comp α β = nat_trans_id F -> nat_trans_comp (#H α) (#H β) = nat_trans_id (pr1 (H F)).
proof: Proof. intro e. intermediate_path (#H (nat_trans_id F)). - rewrite <- e. change ( (# H α) · (# H β) = # H ((α:[C,D']⟦ F, F' ⟧) · (β:[C,D']⟦ F', F ⟧)) ). apply (! functor_comp _ _ _). - apply functor_id_id. apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored UniMath.Bicategories.MonoidalCategories.PointedFunctorsMonoidal UniMath.Bicategories.MonoidalCategories.Actions UniMath.Bicategories.MonoidalCategories.ConstructionOfActions UniMath.Bicategories.MonoidalCategories.ActionOfEndomorphismsInBicat UniMath.Bicategories.MonoidalCategories.ActionBasedStrength UniMath.Bicategories.MonoidalCategories.MonoidalFromBicategory UniMath.Bicategories.MonoidalCategories.ActionBasedStrongFunctorCategory UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.Examples.BicatOfCats UniMath.CategoryTheory.Monoidal.Examples.EndofunctorsMonoidalElementary UniMath.CategoryTheory.Actegories.Examples.ActionOfEndomorphismsInCATElementary UniMath.SubstitutionSystems.Signatures UniMath.SubstitutionSystems.ActionBasedStrengthOnHomsInBicat UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Actegories.Actegories UniMath.CategoryTheory.Actegories.MorphismsOfActegories UniMath.CategoryTheory.Actegories.ConstructionOfActegories UniMath.CategoryTheory.coslicecat UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects UniMath.Bicategories.MonoidalCategories.PointedFunctorsWhiskeredMonoidal UniMath.Bicategories.MonoidalCategories.ActionOfEndomorphismsInBicatWhiskered UniMath.Bicategories.MonoidalCategories.BicatOfActegories