Definition preserves_inserters_left_adjoint_equivalence (HF : preserves_inserters F) : preserves_inserters G. Proof. intros x y f g p Hp. use (has_inserter_ump_left_adjoint_equivalence _ _ _ (Hη x) (Hη y) (Hη p) (psnaturality_of η (inserter_cone_pr1 p)) (psnaturality_of η f) (psnaturality_of η g) (HF _ _ _ _ _ Hp)). abstract (pose (psnaturality_natural η _ _ _ _ (inserter_cone_cell p)) as q ; cbn in q ; rewrite (pstrans_comp_alt η (inserter_cone_pr1 p) g) in q ; rewrite (pstrans_comp_alt η (inserter_cone_pr1 p) f) in q ; rewrite !vassocl in q ; rewrite <- !lwhisker_vcomp ; rewrite <- !rwhisker_vcomp ; rewrite !vassocr ; use vcomp_move_L_Mp ; [ is_iso | ] ; cbn -[psfunctor_comp] ; rewrite !vassocl ; rewrite q ; rewrite !vassocr ; do 6 apply maponpaths_2 ; rewrite !lwhisker_vcomp ; rewrite vcomp_rinv ; rewrite lwhisker_id2 ; apply id2_left). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Core.BicategoryLaws. Require Import UniMath.Bicategories.Core.EquivToAdjequiv. Require Import UniMath.Bicategories.Morphisms.Adjunctions. Require Import UniMath.Bicategories.Morphisms.FullyFaithful. Require Import UniMath.Bicategories.Morphisms.Properties.ContainsAdjEquiv. Require Import UniMath.Bicategories.PseudoFunctors.Display.PseudoFunctorBicat. Require Import UniMath.Bicategories.PseudoFunctors.PseudoFunctor. Import PseudoFunctor.Notations. Require Import UniMath.Bicategories.PseudoFunctors.Preservation.Preservation. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Identity. Require Import UniMath.Bicategories.PseudoFunctors.Examples.Composition. Require Import UniMath.Bicategories.Transformations.PseudoTransformation. Require Import UniMath.Bicategories.Limits.Final. Require Import UniMath.Bicategories.Limits.Products. Require Import UniMath.Bicategories.Limits.ProductEquivalences. Require Import UniMath.Bicategories.Limits.Inserters. Require Import UniMath.Bicategories.Limits.InserterEquivalences. Require Import UniMath.Bicategories.Limits.Equifiers. Require Import UniMath.Bicategories.Limits.EquifierEquivalences. Require Import UniMath.Bicategories.Colimits.Initial. Require Import UniMath.Bicategories.Colimits.Coproducts. Require Import UniMath.Bicategories.Colimits.CoproductEquivalences.