proposition: Lemma presheaf_exponent_is_morphism (P : presheaf L) (m n : nat) (a : (BPO (theory_presheaf L) (plus_1_presheaf P) m : hSet)) (f : stn m → L n) : presheaf_exponent_morphism_data P n (op a f) = op (presheaf_exponent_morphism_data P m a) f.
proof: Proof. refine (op_op P (pr2 a) _ _ @ !_). refine (op_op P (l := S m) (m := m) _ _ _ @ !_). apply (maponpaths (op (pr2 a : (P _ : hSet)))). apply funextfun. intro i. induction (invmap stnweq i) as [i' | i']. - refine (comp_comp _ (f i') _ _ @ !_). refine (pr_comp _ _ _ @ _). refine (!comp_pr _ _ @ !_). apply maponpaths. apply funextfun. intro j. refine (pr_comp _ _ _ @ _). exact (maponpaths _ (homotinvweqweq stnweq _)). - 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