Lemma preCompWithColimOfArrows_subproof {C : precategory} {g : graph} {d1 d2 : diagram g C} (CC1 : ColimCocone d1) (CC2 : ColimCocone d2) (f : ∏ (u : vertex g), C⟦dob d1 u,dob d2 u⟧) (fNat : ∏ u v (e : edge u v), dmor d1 e · f v = f u · dmor d2 e) (x : C) (cc : cocone d2 x) u v (e : edge u v) : dmor d1 e · (f v · coconeIn cc v) = f u · coconeIn cc u. Proof. now rewrite <- (coconeInCommutes cc u v e), !assoc, fNat. 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.FunctorCategory. Require Import UniMath.CategoryTheory.Adjunctions.Core.