proposition: Lemma hom_weq_eq {m n : nat} (G : presheaf_cat L ⟦pow (S n), theory_presheaf L⟧) (l : ((PO n) m : hSet)) : pr1 (φ_adj (pr2 theory_presheaf_exponentiable) G) m l = pr1 G (S m) (pr (stnweq (inr tt)) ,, op l (λ i, pr (stnweq (inl i)))).
proof: Proof. refine (maponpaths (λ x, x _) (presheaf_mor_comp (P'' := plus_1_presheaf (theory_presheaf L)) _ _ _) @ _). refine (maponpaths (λ x, x _) (presheaf_mor_comp (P := BPO _ _) (P'' := theory_presheaf L) _ G _) @ _). refine (maponpaths (pr1 G (S m)) _). apply pathsdirprod. - refine (comp_comp L (pr _) _ _ @ _). refine (pr_comp _ _ _ @ _). refine (maponpaths (λ x, _ x • _) (homotinvweqweq stnweq _) @ _). refine (pr_comp _ _ _ @ _). exact (maponpaths (λ x, _ x) (homotinvweqweq stnweq _)). - refine (op_op (l := S m) (m := S (S m)) _ _ _ _ @ _). refine (op_op (l := m) (m := S m) _ _ _ _ @ _). apply maponpaths. apply funextfun. intro i. refine (pr_comp _ _ _ @ _). refine (maponpaths (λ x, _ x • _) (homotinvweqweq stnweq _) @ _). refine (maponpaths (λ x, x • _) (pr_comp _ _ _) @ _). refine (pr_comp _ _ _ @ _). exact (maponpaths _ (homotinvweqweq stnweq _)). 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