Lemma postCompWithLimArrow {C : precategory} {g : graph} (D : diagram g C) (CC : LimCone D) (c : C) (cc : cone D c) (d : C) (k : C⟦d,c⟧) : k · limArrow CC c cc = limArrow CC d (make_cone (λ u, k · coneOut cc u) (Cone_precompose cc d k)). Proof. apply limArrowUnique. now intro u; rewrite <- assoc, limArrowCommutes. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.Limits.Graphs.Colimits. Import UniMath.CategoryTheory.opp_precat.