Definition identity_is_Pullback_input {a b c : C}{f : C⟦b, a⟧} {g : C⟦c, a⟧} (Pb : Pullback f g) : total2 (fun hk : C⟦lim Pb, lim Pb⟧ => dirprod (hk · PullbackPr1 Pb = PullbackPr1 Pb)(hk · PullbackPr2 Pb = PullbackPr2 Pb)). Proof. exists (identity (lim Pb)). apply make_dirprod; apply id_left. Defined.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Combinatorics.StandardFiniteSets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Limits.Graphs.Limits. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits.