proposition: Lemma form_adjunction_functor_cat_swap (C D : precategory) (E : category) : form_adjunction _ _ (id_functor_cat_swap C D E) (functor_cat_swap_id C D E).
proof: Proof. set (hsE := homset_property E). split; intro F. + apply (nat_trans_eq (functor_category_has_homsets _ _ hsE) (pr1 (pr1 (functor_cat_swap C D E) F)) (pr1 (pr1 (functor_cat_swap C D E) F))). now intro d; apply (nat_trans_eq hsE); intro c; apply id_right. + apply (nat_trans_eq (functor_category_has_homsets _ _ hsE) (pr1 (pr1 (functor_cat_swap D C E) F)) (pr1 (pr1 (functor_cat_swap D C E) F))). now intro d; apply (nat_trans_eq hsE); intro c; apply id_left. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.ProductCategory UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.Products UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Adjunctions.Core