Definition precategory_total_comp (C : precategory_data) : ∏ f g : total_morphisms C, precategory_target C f = precategory_source C g -> total_morphisms C := λ f g e, tpair _ (make_dirprod (pr1 (pr1 f))(pr2 (pr1 g))) ((pr2 f · idtomor _ _ e) · pr2 g).
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Setcategories.