proposition: Lemma presheaf_exponent_induced_morphism_commutes {P P' : presheaf L} (F : presheaf_morphism (BPO (theory_presheaf L) P') P) : F = # (constprod_functor1 (bin_products_presheaf_cat _) (theory_presheaf L)) (presheaf_exponent_induced_morphism F : presheaf_cat L⟦P', plus_1_presheaf P⟧) · presheaf_exponent_morphism P.
proof: Proof. apply (presheaf_morphism_eq F _). intro n. refine (!(presheaf_mor_comp (P := BPO _ _) _ _ _ @ _)). apply funextfun. intro t. refine (maponpaths (λ x, op (P := P) (x t) _) (presheaf_mor_comp (P := BPO _ _) (P'' := plus_1_presheaf P) _ _ _) @ _). refine (!mor_op F _ _ @ _). apply maponpaths. apply pathsdirprod. - refine (pr_comp _ _ _ @ _). refine (maponpaths _ (homotinvweqweq stnweq _) @ _). refine (maponpaths (λ x, pr1 x n t) (id_right (BinProductPr1 _ ((bin_products_presheaf_cat L) _ _))) ). - refine (op_op P' (pr2 t) _ _ @ _). refine (_ @ op_pr _ _). apply maponpaths. apply funextfun. intro i. 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