Definition functor_from_slice_cleaving_nat_trans : functor_from_cleaving (slice_disp_bicat a) (disp_2cells_isaprop_slice a) (disp_univalent_2_1_slice a) (slice_global_cleaving a) (slice_local_iso_cleaving a) f ⟹ slice_fib_to_hom _ _ ∙ core_functor (pre_comp a f) ∙ hom_to_slice_fib _ _. Proof. use make_nat_trans. - exact (λ g, linvunitor_invertible_2cell _). - abstract (intros g₁ g₂ α ; use subtypePath ; [ intro ; apply isaprop_is_invertible_2cell | ] ; cbn in * ; rewrite lwhisker_id2 ; rewrite id2_left, id2_right ; rewrite <- !lwhisker_vcomp ; rewrite !vassocr ; do 3 apply maponpaths_2 ; rewrite !vassocl ; rewrite <- !rwhisker_vcomp ; rewrite !vassocl ; rewrite !(maponpaths (λ z, _ • z) (vassocr _ _ _)) ; rewrite runitor_rwhisker ; rewrite !vassocr ; rewrite lwhisker_vcomp ; rewrite <- vcomp_whisker ; rewrite linvunitor_assoc ; rewrite !vassocl ; apply maponpaths ; rewrite <- lwhisker_lwhisker_rassociator ; rewrite !lwhisker_vcomp ; apply idpath). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Core. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.DisplayedBicats.DispUnivalence. Require Import UniMath.Bicategories.DisplayedBicats.Examples.LaxSlice. Require Import UniMath.Bicategories.DisplayedBicats.Examples.Slice. Require Import UniMath.Bicategories.DisplayedBicats.FiberBicategory. Require Import UniMath.Bicategories.DisplayedBicats.FiberCategory. Require Import UniMath.Bicategories.DisplayedBicats.CleavingOfBicat. Require Import UniMath.Bicategories.DisplayedBicats.ExamplesOfCleavings.SliceCleaving. Require Import UniMath.Bicategories.DisplayedBicats.FiberBicategory.FunctorFromCleaving.