proposition: Lemma invmap_hom_weq_eq {m n : nat} (G : presheaf_cat L ⟦pow n, pr1 theory_presheaf_exponentiable (theory_presheaf L)⟧) (l : ((PO (S n)) m : hSet)) : pr1 (φ_adj_inv (pr2 theory_presheaf_exponentiable) G) m l = pr1 G m (pr2 l) • (extend_tuple pr (pr1 l)).
proof: Proof. refine (maponpaths (λ x, x _) (presheaf_mor_comp (P'' := theory_presheaf L) _ _ _) @ _). refine (maponpaths (λ x, x • _) _ @ maponpaths (λ x, (pr1 G) m (pr2 l) • x) _). - exact (maponpaths (λ x, x _) (presheaf_mor_comp _ G _)). - refine (!extend_tuple_eq _ _). + intro i. exact (!maponpaths _ (homotinvweqweq _ (inl i))). + refine (!_ @ !maponpaths _ (homotinvweqweq _ (inr tt))). exact (maponpaths (λ x, pr1 x _ _) (id_right (BinProductPr1 _ (bin_products_presheaf_cat _ (theory_presheaf L) _))) ). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.Limits UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Examples.Cartesian UniMath.CategoryTheory.DisplayedCats.Examples.Sigma UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Products UniMath.CategoryTheory.Limits.Terminal UniMath.Combinatorics.StandardFiniteSets UniMath.Combinatorics.Tuples UniMath.AlgebraicTheories.AlgebraicTheories UniMath.AlgebraicTheories.AlgebraicTheoryCategory UniMath.AlgebraicTheories.AlgebraicTheoryCategoryCore UniMath.AlgebraicTheories.AlgebraicTheoryMorphisms UniMath.AlgebraicTheories.Examples.EndomorphismTheory UniMath.AlgebraicTheories.Examples.Plus1Presheaf UniMath.AlgebraicTheories.LambdaTheories UniMath.AlgebraicTheories.LambdaTheoryCategory UniMath.AlgebraicTheories.LambdaTheoryCategoryCore UniMath.AlgebraicTheories.LambdaTheoryMorphisms UniMath.AlgebraicTheories.PresheafCategory UniMath.AlgebraicTheories.PresheafCategoryCore UniMath.AlgebraicTheories.PresheafMorphisms UniMath.AlgebraicTheories.Presheaves