Lemma bifunctor_to_functor_to_bifunctor_data {C D E : category} (F : bifunctor C D E) : pr1 (bifunctor_from_functorfromproductcat (bifunctor_to_functorfromproductcat F)) = pr1 F. Proof. repeat (use total2_paths_f). - apply idpath. - repeat (apply funextsec ; intro). apply when_bifunctor_becomes_leftwhiskering. - repeat (apply funextsec ; intro). etrans. { rewrite transportf_const. apply when_bifunctor_becomes_rightwhiskering. } apply idpath. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import  UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.ProductCategory. Import BifunctorNotations. Import BifunctorNotations. Section FunctorsFromProductCategory. Import BifunctorNotations. End FunctorsFromProductCategory. Import BifunctorNotations. Import BifunctorNotations. Import BifunctorNotations.