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. 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.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.Bicategories.MonoidalCategories.PointedFunctorsMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.Actions. Require Import UniMath.Bicategories.MonoidalCategories.ConstructionOfActions. Require Import UniMath.Bicategories.MonoidalCategories.ActionOfEndomorphismsInBicat. Require Import UniMath.Bicategories.MonoidalCategories.ActionBasedStrength. Require Import UniMath.Bicategories.MonoidalCategories.MonoidalFromBicategory. Require Import UniMath.Bicategories.MonoidalCategories.ActionBasedStrongFunctorCategory. Require Import UniMath.Bicategories.Core.Bicat. Require Import UniMath.Bicategories.Core.Examples.BicatOfCats. Require Import UniMath.CategoryTheory.Monoidal.Examples.EndofunctorsMonoidalElementary. Require Import UniMath.CategoryTheory.Actegories.Examples.ActionOfEndomorphismsInCATElementary. Require Import UniMath.SubstitutionSystems.Signatures. Require Import UniMath.SubstitutionSystems.ActionBasedStrengthOnHomsInBicat. Require Import UniMath.CategoryTheory.Monoidal.Categories. Require Import UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal. Require Import UniMath.CategoryTheory.Actegories.Actegories. Require Import UniMath.CategoryTheory.Actegories.MorphismsOfActegories. Require Import UniMath.CategoryTheory.Actegories.ConstructionOfActegories. Require Import UniMath.CategoryTheory.coslicecat. Require Import UniMath.CategoryTheory.Monoidal.Examples.MonoidalPointedObjects. Require Import UniMath.Bicategories.MonoidalCategories.PointedFunctorsWhiskeredMonoidal. Require Import UniMath.Bicategories.MonoidalCategories.ActionOfEndomorphismsInBicatWhiskered. Require Import UniMath.Bicategories.MonoidalCategories.BicatOfActegories. Import Bicat.Notations. Import MonoidalNotations.