proposition: Lemma postCompWithLimOfArrows {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) : limArrow CC1 x cc · limOfArrows CC1 CC2 f fNat = limArrow CC2 x (make_cone (λ u, coneOut cc u · f u) (postCompWithLimOfArrows_subproof CC1 CC2 f fNat x cc)).
proof: Proof. apply limArrowUnique; intro u. now rewrite <- assoc, limOfArrowsOut, assoc, limArrowCommutes. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Propositions UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Limits.Graphs.Colimits