Definition total_category_Binproducts_data (Ps : BinProducts C) (dPs : dispBinProducts Ps) (ccc ddd : total_category D) : ∑ p : total_category D, total_category D ⟦ p, ccc ⟧ × total_category D ⟦ p, ddd ⟧. Proof. induction ccc as [c cc]. induction ddd as [d dd]. - exists (BinProductObject _ (Ps c d) ,, dispBinProductObject (Ps c d) (dPs c d cc dd)). split. + use tpair. * apply BinProductPr1. * apply dispBinProductPr1. + use tpair. * apply BinProductPr2. * apply dispBinProductPr2. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.Equalizers. Require Import UniMath.CategoryTheory.Limits.Products. Require Import UniMath.CategoryTheory.Limits.Coequalizers. Require Import UniMath.CategoryTheory.Limits.BinCoproducts. Require Import UniMath.CategoryTheory.Limits.Coproducts. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total.