Definition preserves_binproduct_equivalence_f (HF : preserves_binproduct F) : preserves_binproduct G. Proof. use (preserves_binproduct_nat_z_iso_f θ). use composition_preserves_binproduct. - use composition_preserves_binproduct. + exact (right_adjoint_preserves_binproduct _ HEC). + exact HF. - exact (right_adjoint_preserves_binproduct _ (adj_equivalence_of_cats_inv HED)). Defined.
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.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Limits.Initial. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.PullbackConstructions. Require Import UniMath.CategoryTheory.Limits.Preservation. Require Import UniMath.CategoryTheory.Limits.EquivalencePreservation. Require Import UniMath.CategoryTheory.Limits.LimitIso. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Equivalences.CompositesAndInverses. Require Import UniMath.CategoryTheory.whiskering. Let prodC : BinProducts C := BinProductsFromPullbacks PBC TC.