Definition all_conjoints_spans : all_conjoints_double_cat spans_double_cat. Proof. intros x y f. simple refine (_ ,, _). - exact (span_conjoint C f). - use make_double_cat_are_conjoints'. + exact (span_conjoint_unit C f). + exact (span_conjoint_counit C f). + abstract (use span_sqr_eq ; refine (transportf_disp_mor2_span _ _ _ _ @ _) ; cbn ; unfold mor_of_comp_span_mor, span_lunitor_mor, span_rinvunitor ; rewrite PullbackArrow_PullbackPr2 ; rewrite !assoc ; rewrite PullbackArrow_PullbackPr2 ; cbn ; apply id_left). + abstract (use span_sqr_eq ; refine (transportf_disp_mor2_span _ _ _ _ @ _) ; cbn ; apply id_left). Defined.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Prelude. Require Import UniMath.CategoryTheory.Core.Setcategories. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness. Require Import UniMath.CategoryTheory.TwoSidedDisplayedCats.Examples.Spans. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Examples.BicatOfUnivCats. Require Import UniMath.Bicategories.DoubleCategories.Basics.DoubleCategoryBasics. Require Import UniMath.Bicategories.DoubleCategories.Core.DoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.UnivalentDoubleCats. Require Import UniMath.Bicategories.DoubleCategories.Core.PseudoDoubleSetCats. Require Import UniMath.Bicategories.DoubleCategories.Core.CompanionsAndConjoints.