Lemma postCompWithLimOfArrows_subproof {C : precategory} {g : graph} {d1 d2 : diagram g C} (CC1 : LimCone d1) (CC2 : LimCone d2) (f : ∏ (u : vertex g), C⟦dob d1 u,dob d2 u⟧) (fNat : ∏ u v (e : edge u v), f u · dmor d2 e = dmor d1 e · f v) (x : C) (cc : cone d1 x) u v (e : edge u v) : (coneOut cc u · f u) · dmor d2 e = coneOut cc v · f v. Proof. now rewrite <- (coneOutCommutes cc u v e), <- assoc, fNat, assoc. Defined.
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.