proposition: Lemma LRB_vcomp_lunitor {x y : B} {f g : R (hom x y)} (α : R(hom x y)⟦f,g⟧) : LRB_lwhisker (η (hom x x) (identity x)) α · LRB_lunitor g = pr1 (LRB_lunitor f) · α.
proof: Proof. use (factor_through_squash _ _ (eso (hom x y) f)). { apply homset_property. } intros [f0 pf]. induction (isotoid _ (pr2 (R (hom x y))) pf). use (factor_through_squash _ _ (eso (hom x y) g)). { apply homset_property. } intros [g0 pg]. induction (isotoid _ (pr2 (R (hom x y))) pg). clear pf pg. etrans. { apply maponpaths. exact (toforallpaths _ _ _ (base_paths _ _ (LRB_lunitor_comm _ _)) g0). } assert (α' : ∑ α0 : (hom x y)⟦f0,g0⟧, #(η_{x,y}) α0 = α). { apply (ff_{x,y} f0 g0). } induction α' as [α0 αp]. induction αp. etrans. 2: apply maponpaths_2, (! prewhisker_LRB_lunitor f0). etrans. 2: { rewrite assoc'. apply maponpaths. apply (functor_comp (η_{x,y})). } etrans. 2: { do 2 apply maponpaths. apply (vcomp_lunitor f0 g0 α0). } etrans. { apply maponpaths, (LRB_lunitor_pre_simpl g0). } etrans. 2: apply maponpaths, pathsinv0, (functor_comp (η_{x,y})). rewrite ! assoc. apply maponpaths_2. etrans. 2: { do 2 apply maponpaths_2. apply pathsinv0, (functor_id (LRB_composition_curry2 x x y (η_{x,y} f0))). } rewrite id_left. refine (_ @ pr21 (LRB_composition_comm x x y) _ _ (id2 (id₁ x) : (hom _ _)⟦_,_⟧ #, α0) @ _) ; cbn. - apply maponpaths_2, maponpaths, maponpaths_2, pathsinv0, (functor_id (η (hom x x))). - do 2 apply maponpaths. apply pathsinv0, lwhisker_hcomp. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.CompositesAndInverses UniMath.CategoryTheory.PrecompEquivalence UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.RezkCompletion UniMath.Bicategories.Core.Bicat UniMath.Bicategories.Core.BicategoryLaws UniMath.Bicategories.Core.Invertible_2cells UniMath.Bicategories.Core.Examples.BicatOfUnivCats UniMath.Bicategories.Core.Univalence UniMath.Bicategories.Core.YonedaLemma UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat UniMath.Bicategories.PseudoFunctors.PseudoFunctor UniMath.Bicategories.PseudoFunctors.Properties UniMath.Bicategories.PseudoFunctors.Examples.Composition